**1. Introduction**

In this chapter, we will discuss the stochastic optimal control problem for jump diffusions. That is, the controlled stochastic system is driven by both Brownian motion and Poisson random measure and the controller wants to minimize/maximize some cost functional subject to the above stated state equation (stochastic control system) over the admissible control set. This kind of stochastic optimal control problems can be encountered naturally when some sudden and rare breaks take place, such as in the practical stock price market. An admissible control is called optimal if it achieves the infimum/supremum of the cost functional and the corresponding state variable and the cost functional are called the optimal trajectory and the value function, respectively.

It is well-known that Pontryagin's maximum principle (MP for short) and Bellman's dynamic programming principle (DPP for short) are the two principal and most commonly used approaches in solving stochastic optimal control problems. In the statement of maximum principle, the necessary condition of optimality is given. This condition is called the maximum condition which is always given by some Hamiltonian function. The Hamiltonian function is defined with respect to the system state variable and some adjoint variables. The equation that the adjoint variables satisfy is called adjoint equation, which is one or two backward stochastic differential equations (BSDEs for short) of [13]'s type. The system which consists of the adjoint equation, the original state equation, and the maximum condition is referred to as a generalized Hamiltonian system. On the other hand, the basic idea of dynamic programming principle is to consider a family of stochastic optimal control problems with different initial time and states and establish relationships among these problems via the so-called Hailton-Jacobi-Bellman (HJB for short) equation, which is a nonlinear second-order partial differential equation (PDE for short). If the HJB equation is solvable, we can obtain an optimal control by taking the maximizer/miminizer of the generalized Hamiltonian function involved in the HJB equation. To a great extent these two approaches have been developed separately and independently during the research in stochastic optimal control problems.

<sup>\*</sup>The main content of this chapter is from the following published article paper: Shi, J.T., & Wu, Z. (2011). Relationship between MP and DPP for the stochastic optimal control problem of jump diffusions. *Applied mathematics and Optimization*, Vol. 63, 151–189.

©2012 Shi, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Shi, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Will-be-set-by-IN-TECH 120 Stochastic Modeling and Control Stochastic Control for Jump Diffusions <sup>1</sup> <sup>3</sup>

Hence, a natural question arises: Are there any relations between these two methods? In fact, the relationship between MP and DPP is essentially the relationship between the adjoint processes and the value function, or the Hamiltonian systems and the HJB equations or even more generally, the relationship between stochastic differential equations (SDEs for short) and PDEs. Such a topic was intuitively discussed by [5], [4] and [9]. However, an important issue in studying the problem is that the derivatives of the value functions are unavoidably involved in these results. In fact, the value functions are usually not necessarily smooth. [19] first obtained the nonsmooth version of the relationship between MP and DPP using the viscosity solution and the second-order adjoint equation. See also the book by [18].

where

defined by

+ � **E** �

⎧ ⎪⎨

⎪⎩

viscosity solutions.

are given in Section 4.

*positive constant.*

**2. Problem statement and preliminaries**

solve the adjoint equation (1).

*F*(*t*, *x*, *u*) := *f*(*t*, *x*, *u*) + *Vt*(*t*, *x*) + �*Vx*(*t*, *x*), *b*(*t*, *x*, *u*)� +

*s*,*y*;*u*¯ (*t*)),

*s*,*y*;*u*¯

*s*,*y*;*u*¯

(*t*))*σ*(*t*, *x*¯

(*t*−) + *c*(*t*, *x*¯

*p*(*t*) = −*Vx*(*t*, *x*¯

*q*(*t*) = −*Vxx*(*t*, *x*¯

*γ*(*t*, ·) = −*Vx*(*t*, *x*¯

1 2 tr�

*V*(*t*, *x* + *c*(*t*, *x*, *u*,*e*)) − *V*(*t*, *x*) − �*Vx*(*t*, *x*), *c*(*t*, *x*, *u*,*e*)�

Moreover, Theorem 2.1 of [7] says that if *<sup>V</sup>*(·, ·) belongs to *<sup>C</sup>*1,3([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*), then the processes

*s*,*y*;*u*¯

However, it seems that the above HJB equation (2) and the relationship (3) lack generality, since they require the value function to be smooth, which is not true even in the simplest case; see Example 3.2 of this chapter. This is an important *gap* in the literature [7]. The aim of this chapter is to bridge this gap by employing the notion of semijets evoked in defining the

The contribution of this chapter is as follows. Firstly, we give some basic properties of the value function and prove that the DPP still holds in our jump diffusion setting. Then we give the corresponding generalized HJB equation which now is a second-order PIDE. Secondly, we investigate the relationship between MP and DPP without assuming the continuous differentiablity of the value function. We obtain the relationship among the adjoint processes, the generalized Hamiltonian and the value function by employing the notions of the set-valued semijets evoked in defining the viscosity solutions, which is now interpreted as a set inclusion form among subjet, superjet of the value function, set contain adjoint processes and some "G-function" (see the definition in Section 2). It is worth to pointed out that the controlled jump diffusions bring much technique difficulty to obtain the above results. In fact, the solution of the control system is not continuous with jump diffusions. We overcome these difficulty and get the desired results in this chapter which have wide applicable background. The rest of this chapter is organized as follows. In Section 2, for stochastic optimal control problem of jump diffusions, we give some basic properties of the value function and then set out the corresponding DPP and MP, respectively. In Section 3, the relationship between MP and DPP is proved using the notion of viscosity solutions of PIDEs. Some concluding remarks

*Throughout this chapter, we denote by* **R***<sup>n</sup> the space of n-dimensional Euclidean space, by* **R***n*×*<sup>d</sup> the space of matrices with order n* <sup>×</sup> *d, by* <sup>S</sup>*<sup>n</sup> the space of symmetric matrices with order n* <sup>×</sup> *n.* �·, ·� *and* |·| *denote the scalar product and norm in the Euclidean space, respectively.* � *appearing in the superscripts denotes the transpose of a matrix. a* ∨ *b denotes* max{*a*, *b*}*. C always denotes some*

*s*,*y*;*u*¯

(*t*), *u*¯(*t*)),

(*t*−), *u*¯(*t*)) + *Vx*(*t*, *x*¯

*Vxx*(*t*, *<sup>x</sup>*)*σ*(*t*, *<sup>x</sup>*, *<sup>u</sup>*)*σ*(*t*, *<sup>x</sup>*, *<sup>u</sup>*)��

Stochastic Control for Jump Diff usions 121

� *π*(*de*).

*s*,*y*;*u*¯

(*t*−)),

(3)

The aim of this chapter is to establish the relationship between MP and DPP within the framework of viscosity solutions in the jump diffusion setting. In this case, the state trajectory is described by a stochastic differential equation with Poisson jumps (SDEP for short). That is to say, the system noise (or the uncertainty of the problem) comes from a Brownian motion and a Poisson random measure. See [15] for theory and applications of this kind of equations. [16] proved the general MP where the control variable is allowed into both diffusion and jump coefficients. HJB equation for optimal control of jump diffusions can be seen in [12], which here is a second-order partial integral-differential equation (PIDE for short). [7] gave a sufficient MP by employing Arrow's generalization of the Mangasarian sufficient condition to the jump diffusion setting. Moreover, on the assumption that the value function is smooth, they showed the adjoint processes' connections to the value function. Let us state some results of [7] in detail with a slight modification to adapt to our setting.

A sufficient MP was proved to say that, for any admissible pair (*x*¯*s*,*y*;*u*¯ (·), *u*¯(·)), if there exists an adapted solution (*p*(·), *q*(·), *γ*(·, ·)) of the following adjoint equtaion

$$\begin{cases} -dp(t) = H\_{\mathcal{X}}(t, \bar{\mathbf{x}}^{\mathcal{Y}, \mathcal{Y}}(t), \bar{u}(t), p(t), q(t), \gamma(t\_{\prime} \cdot))dt - q(t)dW(t) \\ \qquad - \int\_{\mathcal{E}} \gamma(t, e)\tilde{N}(dedt), \; t \in [0, T], \\\ p(T) = -h\_{\mathcal{X}}(\bar{\mathbf{x}}^{\mathcal{Y}, \mathcal{Y}}(T)), \end{cases} \tag{1}$$

which is a BSDE with Poisson jumps (BSDEP for short) such that

$$H(t, \bar{\mathfrak{x}}^{\mathbf{s}, \mathbf{y}; \mathfrak{R}}(t), \bar{\mathfrak{u}}(t), p(t), q(t), \gamma(t, \cdot)) = \sup\_{\boldsymbol{\mu} \in \bar{\mathbf{U}}} H(t, \bar{\mathfrak{x}}^{\mathbf{s}, \mathbf{y}; \mathfrak{R}}(t), \boldsymbol{\mu}, p(t), q(t), \gamma(t, \cdot))$$

for all *t* ∈ [0, *T*] and that

$$\hat{H}(\mathbf{x}) := \max\_{\boldsymbol{\mu} \in \mathbf{U}} H(t, \mathbf{x}, \boldsymbol{\mu}, \boldsymbol{p}(t), \boldsymbol{q}(t), \boldsymbol{\gamma}(t, \cdot))$$

exists and is a concave function of *<sup>x</sup>* for all *<sup>t</sup>* <sup>∈</sup> [0, *<sup>T</sup>*], then (*x*¯*s*,*y*;*u*¯ (·), *u*¯(·)) is an optimal pair. In the above, the Hamiltonian function *<sup>H</sup>* : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>R</sup>***n*×*<sup>d</sup>* × L2(**E**, <sup>B</sup>(**E**), *<sup>π</sup>*; **<sup>R</sup>***n*) <sup>→</sup> **R** is defined as

$$H(t, \mathbf{x}, u, p, q, \gamma(\cdot)) := f(t, \mathbf{x}, u) + \langle p, b(t, \mathbf{x}, u) \rangle + \text{tr}\{\sigma(t, \mathbf{x}, u)^{\top} q\} + \int\_{\mathbf{E}} \langle \gamma(\varepsilon), \varepsilon(t, \mathbf{x}, u, \varepsilon) \rangle \pi(d\varepsilon).$$

On the other hand, a DPP asserted that if the value function *<sup>V</sup>*(·, ·) belongs to *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*), then it satisfies the following HJB equation

$$\sup\_{\mu \in \mathbf{U}} F(t, \mathbf{x}, \mu) = F(t, \mathbf{x}, \overline{\mu}(t)) = 0 \tag{2}$$

where

2 Will-be-set-by-IN-TECH

Hence, a natural question arises: Are there any relations between these two methods? In fact, the relationship between MP and DPP is essentially the relationship between the adjoint processes and the value function, or the Hamiltonian systems and the HJB equations or even more generally, the relationship between stochastic differential equations (SDEs for short) and PDEs. Such a topic was intuitively discussed by [5], [4] and [9]. However, an important issue in studying the problem is that the derivatives of the value functions are unavoidably involved in these results. In fact, the value functions are usually not necessarily smooth. [19] first obtained the nonsmooth version of the relationship between MP and DPP using the viscosity

The aim of this chapter is to establish the relationship between MP and DPP within the framework of viscosity solutions in the jump diffusion setting. In this case, the state trajectory is described by a stochastic differential equation with Poisson jumps (SDEP for short). That is to say, the system noise (or the uncertainty of the problem) comes from a Brownian motion and a Poisson random measure. See [15] for theory and applications of this kind of equations. [16] proved the general MP where the control variable is allowed into both diffusion and jump coefficients. HJB equation for optimal control of jump diffusions can be seen in [12], which here is a second-order partial integral-differential equation (PIDE for short). [7] gave a sufficient MP by employing Arrow's generalization of the Mangasarian sufficient condition to the jump diffusion setting. Moreover, on the assumption that the value function is smooth, they showed the adjoint processes' connections to the value function. Let us state some results

(*t*), *u*¯(*t*), *p*(*t*), *q*(*t*), *γ*(*t*, ·))*dt* − *q*(*t*)*dW*(*t*)

*s*,*y*;*u*¯

(*t*), *u*, *p*(*t*), *q*(*t*), *γ*(*t*, ·))

(·), *u*¯(·)) is an optimal pair. In

�*γ*(*e*), *c*(*t*, *x*, *u*,*e*)�*π*(*de*).

(·), *u*¯(·)), if there exists

(1)

solution and the second-order adjoint equation. See also the book by [18].

of [7] in detail with a slight modification to adapt to our setting.

−*dp*(*t*) = *Hx*(*t*, *x*¯

− � **E**

*p*(*T*) = −*hx*(*x*¯

⎧ ⎪⎪⎪⎨

⎪⎪⎪⎩

*s*,*y*;*u*¯

*H*(*t*, *x*¯

for all *t* ∈ [0, *T*] and that

**R** is defined as

A sufficient MP was proved to say that, for any admissible pair (*x*¯*s*,*y*;*u*¯

an adapted solution (*p*(·), *q*(·), *γ*(·, ·)) of the following adjoint equtaion

*s*,*y*;*u*¯

*s*,*y*;*u*¯ (*T*)),

(*t*), *u*¯(*t*), *p*(*t*), *q*(*t*), *γ*(*t*, ·)) = sup

*H*ˆ (*x*) := max

exists and is a concave function of *<sup>x</sup>* for all *<sup>t</sup>* <sup>∈</sup> [0, *<sup>T</sup>*], then (*x*¯*s*,*y*;*u*¯

sup *u*∈**U**

*H*(*t*, *x*, *u*, *p*, *q*, *γ*(·)) := *f*(*t*, *x*, *u*) + �*p*, *b*(*t*, *x*, *u*)� + tr

then it satisfies the following HJB equation

*u*∈**U**

which is a BSDE with Poisson jumps (BSDEP for short) such that

*<sup>γ</sup>*(*t*,*e*)*N*˜ (*dedt*), *<sup>t</sup>* <sup>∈</sup> [0, *<sup>T</sup>*],

*u*∈**U**

the above, the Hamiltonian function *<sup>H</sup>* : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>R</sup>***n*×*<sup>d</sup>* × L2(**E**, <sup>B</sup>(**E**), *<sup>π</sup>*; **<sup>R</sup>***n*) <sup>→</sup>

On the other hand, a DPP asserted that if the value function *<sup>V</sup>*(·, ·) belongs to *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*),

*H*(*t*, *x*¯

*H*(*t*, *x*, *u*, *p*(*t*), *q*(*t*), *γ*(*t*, ·))

�

*σ*(*t*, *x*, *u*)�*q*

� + � **E**

*F*(*t*, *x*, *u*) = *F*(*t*, *x*, *u*¯(*t*)) = 0 (2)

$$F(t, \mathbf{x}, \boldsymbol{u}) := f(t, \mathbf{x}, \boldsymbol{u}) + V\_{\mathrm{l}}(t, \mathbf{x}) + \langle V\_{\mathrm{x}}(t, \mathbf{x}), b(t, \mathbf{x}, \boldsymbol{u}) \rangle + \frac{1}{2} \mathrm{tr} \Big\{ V\_{\mathrm{XX}}(t, \mathbf{x}) \sigma(t, \mathbf{x}, \boldsymbol{u}) \sigma(t, \mathbf{x}, \boldsymbol{u})^{\top} \Big\},$$

$$+ \int\_{\mathbf{E}} \left[ V(t, \mathbf{x} + c(t, \mathbf{x}, \boldsymbol{u}, \boldsymbol{e})) - V(t, \mathbf{x}) - \langle V\_{\mathrm{x}}(t, \mathbf{x}), c(t, \mathbf{x}, \boldsymbol{u}, \boldsymbol{e}) \rangle \right] \pi(d\mathbf{e}).$$

Moreover, Theorem 2.1 of [7] says that if *<sup>V</sup>*(·, ·) belongs to *<sup>C</sup>*1,3([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*), then the processes defined by

$$\begin{cases} \quad p(t) = -V\_{\mathcal{X}}(t, \boldsymbol{\tilde{x}}^{\boldsymbol{\mathcal{Y}}, \boldsymbol{\mathcal{Y}}}(t)), \\ \quad q(t) = -V\_{\mathcal{X}\mathcal{X}}(t, \boldsymbol{\tilde{x}}^{\boldsymbol{\mathcal{Y}}, \boldsymbol{\mathcal{Y}}\boldsymbol{\mathcal{R}}}(t)) \sigma(t, \boldsymbol{\tilde{x}}^{\boldsymbol{\mathcal{Y}}, \boldsymbol{\mathcal{Y}}\boldsymbol{\mathcal{R}}}(t), \boldsymbol{\tilde{u}}(t)), \\ \quad \gamma(t, \cdot) = -V\_{\mathcal{X}}(t, \boldsymbol{\tilde{x}}^{\boldsymbol{\mathcal{Y}}, \boldsymbol{\mathcal{Y}}\boldsymbol{\mathcal{R}}}(t -) + c(t, \boldsymbol{\tilde{x}}^{\boldsymbol{\mathcal{Y}}, \boldsymbol{\mathcal{Y}}\boldsymbol{\mathcal{R}}}(t -), \boldsymbol{\tilde{u}}(t)) + V\_{\mathcal{X}}(t, \boldsymbol{\tilde{x}}^{\boldsymbol{\mathcal{Y}}, \boldsymbol{\mathcal{Y}}\boldsymbol{\mathcal{R}}}(t -)), \end{cases} \tag{3}$$

solve the adjoint equation (1).

However, it seems that the above HJB equation (2) and the relationship (3) lack generality, since they require the value function to be smooth, which is not true even in the simplest case; see Example 3.2 of this chapter. This is an important *gap* in the literature [7]. The aim of this chapter is to bridge this gap by employing the notion of semijets evoked in defining the viscosity solutions.

The contribution of this chapter is as follows. Firstly, we give some basic properties of the value function and prove that the DPP still holds in our jump diffusion setting. Then we give the corresponding generalized HJB equation which now is a second-order PIDE. Secondly, we investigate the relationship between MP and DPP without assuming the continuous differentiablity of the value function. We obtain the relationship among the adjoint processes, the generalized Hamiltonian and the value function by employing the notions of the set-valued semijets evoked in defining the viscosity solutions, which is now interpreted as a set inclusion form among subjet, superjet of the value function, set contain adjoint processes and some "G-function" (see the definition in Section 2). It is worth to pointed out that the controlled jump diffusions bring much technique difficulty to obtain the above results. In fact, the solution of the control system is not continuous with jump diffusions. We overcome these difficulty and get the desired results in this chapter which have wide applicable background.

The rest of this chapter is organized as follows. In Section 2, for stochastic optimal control problem of jump diffusions, we give some basic properties of the value function and then set out the corresponding DPP and MP, respectively. In Section 3, the relationship between MP and DPP is proved using the notion of viscosity solutions of PIDEs. Some concluding remarks are given in Section 4.

## **2. Problem statement and preliminaries**

*Throughout this chapter, we denote by* **R***<sup>n</sup> the space of n-dimensional Euclidean space, by* **R***n*×*<sup>d</sup> the space of matrices with order n* <sup>×</sup> *d, by* <sup>S</sup>*<sup>n</sup> the space of symmetric matrices with order n* <sup>×</sup> *n.* �·, ·� *and* |·| *denote the scalar product and norm in the Euclidean space, respectively.* � *appearing in the superscripts denotes the transpose of a matrix. a* ∨ *b denotes* max{*a*, *b*}*. C always denotes some positive constant.*

Let **<sup>E</sup>** <sup>⊂</sup> **<sup>R</sup>***<sup>l</sup>* be a nonempty Borel set equipped with its Borel field <sup>B</sup>(**E**). Let *<sup>π</sup>*(·) be a bounded positive measure on (**E**, <sup>B</sup>(**E**)). We denote by <sup>L</sup>2(**E**, <sup>B</sup>(**E**), *<sup>π</sup>*; **<sup>R</sup>***n*) or <sup>L</sup><sup>2</sup> the set of square integrable functions *<sup>k</sup>*(·) : **<sup>E</sup>** <sup>→</sup> **<sup>R</sup>***<sup>n</sup>* such that ||*k*(·)||<sup>2</sup> <sup>L</sup><sup>2</sup> :<sup>=</sup> � **<sup>E</sup>** |*k*(*e*)| <sup>2</sup>*π*(*de*) < ∞.

where *<sup>f</sup>* : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>→</sup> **<sup>R</sup>**, *<sup>h</sup>* : **<sup>R</sup>***<sup>n</sup>* <sup>→</sup> **<sup>R</sup>** are given functions. For given (*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n*, the *stochastic optimal control problem* is to minimize (5) subject to (4) over U[*s*, *T*]. An admissible

*<sup>V</sup>*(*s*, *<sup>y</sup>*) = inf *<sup>u</sup>*(·)∈U[*s*,*T*]) *<sup>J</sup>*(*s*, *<sup>y</sup>*; *<sup>u</sup>*(·)), <sup>∀</sup>(*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n*,

In this section, we first give some basic properties of the value function. Then we prove that the DPP still holds and introduce the generalized HJB equation and the generalized Hamiltonian function. The idea of proof is originated from [17], [18] while on some different assumptions. Then we introduce the Hamiltonian function, the H-function and adjoint

**(H1)** *b*, *σ*, *c* are uniformly continuous in (*t*, *x*, *u*). There exists a constant *C* > 0, such that

<sup>|</sup>*b*(*t*, *<sup>x</sup>*, *<sup>u</sup>*)<sup>|</sup> <sup>+</sup> <sup>|</sup>*σ*(*t*, *<sup>x</sup>*, *<sup>u</sup>*)<sup>|</sup> <sup>+</sup> ||*c*(*t*, *<sup>x</sup>*, *<sup>u</sup>*, ·)||L<sup>2</sup> <sup>≤</sup> *<sup>C</sup>*(<sup>1</sup> <sup>+</sup> <sup>|</sup>*x*|), <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, *<sup>T</sup>*], *<sup>x</sup>*, *<sup>x</sup>*<sup>ˆ</sup> <sup>∈</sup> **<sup>R</sup>***n*, *<sup>u</sup>*, *<sup>u</sup>*<sup>ˆ</sup> <sup>∈</sup> **<sup>U</sup>**.

**(H2)** *f* , *h* are uniformly continuous in (*t*, *x*, *u*). There exists a constant *C* > 0 and an increasing, continuous function *ω*¯ <sup>0</sup> : [0, ∞) × [0, ∞) → [0, ∞) which satisfies *ω*¯ <sup>0</sup>(*r*, 0) = 0, ∀*r* ≥ 0, such

� <sup>|</sup> *<sup>f</sup>*(*t*, *<sup>x</sup>*, *<sup>u</sup>*) <sup>−</sup> *<sup>f</sup>*(*t*, *<sup>x</sup>*ˆ, *<sup>u</sup>*)<sup>|</sup> <sup>+</sup> <sup>|</sup>*h*(*x*) <sup>−</sup> *<sup>h</sup>*(*x*ˆ)| ≤ *<sup>ω</sup>*¯ <sup>0</sup>(|*x*|∨|*x*ˆ|, <sup>|</sup>*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*ˆ|),

Obviously, under assumption **(H1)**, for any *u*(·) ∈ U[*s*, *T*], SDEP (4) admits a unique solution

**Remark 2.1** We point out that assumption **(H2)** on *f* , *h* allows them to have polynomial (in particular, quadratic) growth in *x*, provided that *b*, *σ*, *c* have linear growth. A typical example is the stochastic linear quadratic (LQ for short) problem. Note that **(H2)** is different from assumptions (2.5), pp. 4 of [14] and assumption **(S2)'**, pp. 248 of [18], where global Lipschitz

**Lemma 2.1** *Let* **(H1)** *hold. For k* = 2, 4*, there exists C* > 0 *such that for any* 0 ≤ *s*,*s*ˆ ≤ *T*, *y*, *y*ˆ ∈

*<sup>k</sup>*)(*<sup>T</sup>* <sup>−</sup> *<sup>s</sup>*)

≤ *C*(1 + |*y*|

� �*s* − *s*ˆ � � *k*

*k*

*<sup>k</sup>*)(*<sup>T</sup>* <sup>−</sup> *<sup>s</sup>*)

*<sup>k</sup>*), *<sup>t</sup>* <sup>∈</sup> [*s*, *<sup>T</sup>*], (7)

*k*

<sup>2</sup> , *t* ∈ [*s*, *T*], (8)

, *t* ∈ [*s*, *T*], (10)

<sup>2</sup> , *t* ∈ [*s* ∨ *s*ˆ, *T*]. (11)

<sup>2</sup> , (9)

*<sup>k</sup>* <sup>≤</sup> *<sup>C</sup>*(<sup>1</sup> <sup>+</sup> <sup>|</sup>*y*<sup>|</sup>

� � �*k*

� � *<sup>k</sup>* <sup>≤</sup> *<sup>C</sup>* � �*y* − *y*ˆ � � *k*

*<sup>k</sup>* <sup>≤</sup> *<sup>C</sup>*(<sup>1</sup> <sup>+</sup> <sup>|</sup>*y*|)

*<sup>k</sup>* <sup>≤</sup> *<sup>C</sup>*(<sup>1</sup> <sup>+</sup> <sup>|</sup>*y*<sup>|</sup>

<sup>|</sup> *<sup>f</sup>*(*t*, 0, *<sup>u</sup>*)|, <sup>|</sup>*h*(0)| ≤ *<sup>C</sup>*, <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, *<sup>T</sup>*], *<sup>x</sup>*, *<sup>x</sup>*<sup>ˆ</sup> <sup>∈</sup> **<sup>R</sup>***n*, *<sup>u</sup>* <sup>∈</sup> **<sup>U</sup>**.

(·), *u*¯(·)) is called *optimal* if *u*¯(·) achieves the infimum of *J*(*s*, *y*; *u*(·)) over U[*s*, *T*]. We

(6)

Stochastic Control for Jump Diff usions 123

pair (*x*¯*s*,*y*;*u*¯

⎧ ⎪⎨

⎪⎩

that

*<sup>x</sup>s*,*y*;*u*(·) (see [10]).

**<sup>R</sup>***n*, *<sup>u</sup>*(·) ∈ U[*s*, *<sup>T</sup>*]*,*

We need the following lemma.

define the *value function V* : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>→</sup> **<sup>R</sup>** as

*<sup>V</sup>*(*T*, *<sup>y</sup>*) = *<sup>h</sup>*(*y*), <sup>∀</sup>*<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>***n*.

processes, then give the MP which is a special case of [16].


We first discuss the DPP and make the following assumptions.

condition is both imposed thus *f* , *h* are global linear growth in *x*.

**E** � �*xs*,*y*,*u*(*t*) � �

�*xs*,*y*,*u*(*t*) <sup>−</sup> *<sup>y</sup>*

�*xs*,*y*,*u*(*t*) <sup>−</sup> *<sup>x</sup>s*ˆ,*y*,*u*(*t*)

�

�*xs*,*y*,*u*(*t*) <sup>−</sup> *<sup>x</sup>s*,*y*ˆ,*u*(*t*)

� �

�*xs*,*y*,*u*(*t*) <sup>−</sup> *<sup>y</sup>*

� �

**E** �

> **E** � sup *s*≤*t*≤*T*

**E** �

**E** �

<sup>+</sup> ||*c*(*t*, *<sup>x</sup>*, *<sup>u</sup>*, ·) <sup>−</sup> *<sup>c</sup>*(*t*, *<sup>x</sup>*ˆ, *<sup>u</sup>*ˆ, ·)||L<sup>2</sup> <sup>≤</sup> *<sup>C</sup>*(|*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*ˆ<sup>|</sup> <sup>+</sup> <sup>|</sup>*<sup>u</sup>* <sup>−</sup> *<sup>u</sup>*ˆ|),

⎧ ⎨ ⎩

Let *T* > 0 and let (Ω, F, **P**) be a complete probability space, equipped with a *d*-dimensional standard Brownian motion {*W*(*t*)}0≤*t*≤*<sup>T</sup>* and a Poisson random measure *N*(·, ·) independent of *<sup>W</sup>*(·) with the intensity measure *<sup>N</sup>*<sup>ˆ</sup> (*dedt*) = *<sup>π</sup>*(*de*)*dt*. We write *<sup>N</sup>*˜ (*dedt*) :<sup>=</sup> *<sup>N</sup>*(*dedt*) <sup>−</sup> *π*(*de*)*dt* for the compensated Poisson martingale measure.

For a given *<sup>s</sup>* <sup>∈</sup> [0, *<sup>T</sup>*), we suppose the filtration {F*<sup>s</sup> <sup>t</sup>* }*s*≤*t*≤*<sup>T</sup>* is generated as the following

$$\mathcal{F}\_t^s := \sigma \left\{ N(\mathbf{A} \times (s, r]); s \le r \le t, \mathbf{A} \in \mathcal{B}(\mathbf{E}) \right\} \bigvee \sigma \left\{ \mathcal{W}(r) - \mathcal{W}(s); s \le r \le t \right\} \bigvee \mathcal{N}\_{\tau}$$

where N contains all **P**-null sets in F and *σ*<sup>1</sup> � *<sup>σ</sup>*<sup>2</sup> denotes the *<sup>σ</sup>*-field generated by *<sup>σ</sup>*<sup>1</sup> ∪ *<sup>σ</sup>*2. In particular, if *<sup>s</sup>* <sup>=</sup> 0 we write <sup>F</sup>*<sup>t</sup>* ≡ F*<sup>s</sup> t* .

Let **<sup>U</sup>** be a nonempty Borel subset of **<sup>R</sup>***k*. For any initial time *<sup>s</sup>* <sup>∈</sup> [0, *<sup>T</sup>*) and initial state *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>***n*, we consider the following stochastic control system which is called a *controlled jump diffusion process*

$$\begin{cases} dx^{s,y;\mu}(t) = b(t, x^{s,y;\mu}(t), \mu(t))dt + \sigma(t, x^{s,y;\mu}(t), \mu(t))dW(t) \\ \qquad + \int\_{\mathbb{E}} c(t, x^{s,y;\mu}(t-), \mu(t), e)\tilde{N}(de dt), \quad t \in (s, T], \\\ x^{s,y;\mu}(s) = y. \end{cases} \tag{4}$$

Here *<sup>b</sup>* : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>→</sup> **<sup>R</sup>***n*, *<sup>σ</sup>* : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>→</sup> **<sup>R</sup>***n*×*d*, *<sup>c</sup>* : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>×</sup> **<sup>E</sup>** <sup>→</sup> **<sup>R</sup>***<sup>n</sup>* are given functions.

For a given *<sup>s</sup>* <sup>∈</sup> [0, *<sup>T</sup>*), we denote by <sup>U</sup>[*s*, *<sup>T</sup>*] the set of **<sup>U</sup>**-valued <sup>F</sup>*<sup>s</sup> <sup>t</sup>* -predictable processes. For given *<sup>u</sup>*(·) ∈ U[*s*, *<sup>T</sup>*] and *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>***n*, a **<sup>R</sup>***n*-valued process *<sup>x</sup>s*,*y*;*u*(·) is called a solution of (4) if it is an <sup>F</sup>*<sup>s</sup> <sup>t</sup>* -adapted RCLL (i.e., right-continuous with left-hand limits) process such that (4) holds. It is called the *state trajectory* corresponding to the control *u*(·) ∈ U[*s*, *T*] with initial state *y*. We refer to such *<sup>u</sup>*(·) ∈ U[*s*, *<sup>T</sup>*] as an *admissible control* and (*xs*,*y*;*u*(·), *<sup>u</sup>*(·)) as an *admissible pair*. For any *s* ∈ [0, *T*), we introduce the following notations.

$$L^{2}(\Omega, \mathcal{F}\_{T}^{\mathrm{s}}; \mathbb{R}^{n}) := \left\{ \mathbb{R}^{n}\text{-valued } \mathcal{F}\_{T}^{\mathrm{s}}\text{-measurable random variables } \mathbb{S}; \,\,\mathbb{E}|\mathcal{G}|^{2} < \infty \right\},$$

$$L^{2}\_{\mathcal{F}}([s, T]; \mathbb{R}^{n}) := \left\{ \mathbb{R}^{n}\text{-valued } \mathcal{F}\_{t}^{\mathrm{s}}\text{-adapted processes } \varphi(t); \,\,\mathbb{E}\int\_{s}^{T}|\boldsymbol{\varphi}(t)|^{2}dt < \infty \right\},$$

$$L^{2}\_{\mathcal{F}, \boldsymbol{\eta}}([s, T]; \mathbb{R}^{n}) := \left\{ \mathbb{R}^{n}\text{-valued } \mathcal{F}\_{t}^{\mathrm{s}}\text{-particle processes } \boldsymbol{\phi}(t); \,\,\mathbb{E}\int\_{s}^{T}|\boldsymbol{\phi}(t)|^{2}dt < \infty \right\},$$

$$\mathbb{F}\_{p}^{2}([s, T]; \mathbb{R}^{n}) := \left\{ \mathbb{R}^{n}\text{-valued } \mathcal{F}\_{t}^{\mathrm{s}}\text{-particle vector processes } \psi(t, e) \text{ defined on}$$

$$\Omega \times [0, T] \times \mathbb{E}; \,\,\mathbb{E} \int\_{s}^{T} \int\_{\mathbf{E}}|\psi(t, e)|^{2}\pi(de)dt < \infty \right\}.$$

We consider the following *cost functional*

$$J(s, y; u(\cdot)) = \mathbb{E}\left[\int\_s^T f(t, \mathbf{x}^{s, y; \mu}(t), u(t))dt + h(\mathbf{x}^{s, y; \mu}(T))\right],\tag{5}$$

where *<sup>f</sup>* : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>→</sup> **<sup>R</sup>**, *<sup>h</sup>* : **<sup>R</sup>***<sup>n</sup>* <sup>→</sup> **<sup>R</sup>** are given functions. For given (*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n*, the *stochastic optimal control problem* is to minimize (5) subject to (4) over U[*s*, *T*]. An admissible pair (*x*¯*s*,*y*;*u*¯ (·), *u*¯(·)) is called *optimal* if *u*¯(·) achieves the infimum of *J*(*s*, *y*; *u*(·)) over U[*s*, *T*]. We define the *value function V* : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>→</sup> **<sup>R</sup>** as

$$\begin{cases} \begin{aligned} V(s,y) &= \inf\_{u(\cdot) \in \mathcal{U}[s,T]} J(s,y;u(\cdot)), \quad \forall (s,y) \in [0,T) \times \mathbf{R}^n, \\ V(T,y) &= h(y), \quad \forall y \in \mathbf{R}^n. \end{aligned} \end{cases} \tag{6}$$

In this section, we first give some basic properties of the value function. Then we prove that the DPP still holds and introduce the generalized HJB equation and the generalized Hamiltonian function. The idea of proof is originated from [17], [18] while on some different assumptions. Then we introduce the Hamiltonian function, the H-function and adjoint processes, then give the MP which is a special case of [16].

We first discuss the DPP and make the following assumptions.

**(H1)** *b*, *σ*, *c* are uniformly continuous in (*t*, *x*, *u*). There exists a constant *C* > 0, such that

$$\begin{cases} \left| b(t, \mathbf{x}, \boldsymbol{u}) - b(t, \mathbf{f}, \hat{\mathbf{u}}) \right| + \left| \sigma(t, \mathbf{x}, \boldsymbol{u}) - \sigma(t, \mathbf{f}, \hat{\mathbf{u}}) \right| \\ \quad + \left| \left| c(t, \mathbf{x}, \boldsymbol{u}\_{\prime} \cdot) - c(t, \mathbf{f}, \hat{\mathbf{u}}\_{\prime} \cdot) \right| \right|\_{\mathcal{L}^{2}} \leq \mathbb{C} (|\mathbf{x} - \hat{\mathbf{x}}| + |\boldsymbol{u} - \hat{\mathbf{u}}|)\_{\prime} \\ \left| b(t, \mathbf{x}, \boldsymbol{u}) \right| + \left| \sigma(t, \mathbf{x}, \boldsymbol{u}) \right| + \left| |c(t, \mathbf{x}, \boldsymbol{u}\_{\prime} \cdot) \right| \right|\_{\mathcal{L}^{2}} \leq \mathbb{C} (1 + |\mathbf{x}|)\_{\prime} \quad \forall t \in [0, T], \mathbf{x}, \hat{\mathbf{x}} \in \mathbf{R}^{n}, \boldsymbol{u}, \hat{\mathbf{u}} \in \mathbf{U}. \end{cases}$$

**(H2)** *f* , *h* are uniformly continuous in (*t*, *x*, *u*). There exists a constant *C* > 0 and an increasing, continuous function *ω*¯ <sup>0</sup> : [0, ∞) × [0, ∞) → [0, ∞) which satisfies *ω*¯ <sup>0</sup>(*r*, 0) = 0, ∀*r* ≥ 0, such that

$$\begin{cases} |f(t, \mathbf{x}, \boldsymbol{\mu}) - f(t, \mathbf{\hat{x}}, \boldsymbol{\mu})| + |h(\mathbf{x}) - h(\mathbf{\hat{x}})| \le \bar{\omega}\_0 (|\mathbf{x}| \vee |\mathbf{\hat{x}}| , |\mathbf{x} - \mathbf{\hat{x}}|),\\ |f(t, 0, \boldsymbol{\mu})| \, |h(\mathbf{0})| \le \mathbf{C}, \quad \forall t \in [0, T], \boldsymbol{\mu}, \boldsymbol{\p} \in \mathbf{R}^n, \boldsymbol{\mu} \in \mathbf{U}.\end{cases}$$

Obviously, under assumption **(H1)**, for any *u*(·) ∈ U[*s*, *T*], SDEP (4) admits a unique solution *<sup>x</sup>s*,*y*;*u*(·) (see [10]).

**Remark 2.1** We point out that assumption **(H2)** on *f* , *h* allows them to have polynomial (in particular, quadratic) growth in *x*, provided that *b*, *σ*, *c* have linear growth. A typical example is the stochastic linear quadratic (LQ for short) problem. Note that **(H2)** is different from assumptions (2.5), pp. 4 of [14] and assumption **(S2)'**, pp. 248 of [18], where global Lipschitz condition is both imposed thus *f* , *h* are global linear growth in *x*.

We need the following lemma.

4 Will-be-set-by-IN-TECH

Let **<sup>E</sup>** <sup>⊂</sup> **<sup>R</sup>***<sup>l</sup>* be a nonempty Borel set equipped with its Borel field <sup>B</sup>(**E**). Let *<sup>π</sup>*(·) be a bounded positive measure on (**E**, <sup>B</sup>(**E**)). We denote by <sup>L</sup>2(**E**, <sup>B</sup>(**E**), *<sup>π</sup>*; **<sup>R</sup>***n*) or <sup>L</sup><sup>2</sup> the set of

Let *T* > 0 and let (Ω, F, **P**) be a complete probability space, equipped with a *d*-dimensional standard Brownian motion {*W*(*t*)}0≤*t*≤*<sup>T</sup>* and a Poisson random measure *N*(·, ·) independent of *<sup>W</sup>*(·) with the intensity measure *<sup>N</sup>*<sup>ˆ</sup> (*dedt*) = *<sup>π</sup>*(*de*)*dt*. We write *<sup>N</sup>*˜ (*dedt*) :<sup>=</sup> *<sup>N</sup>*(*dedt*) <sup>−</sup>

Let **<sup>U</sup>** be a nonempty Borel subset of **<sup>R</sup>***k*. For any initial time *<sup>s</sup>* <sup>∈</sup> [0, *<sup>T</sup>*) and initial state *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>***n*, we consider the following stochastic control system which is called a *controlled jump diffusion*

*dxs*,*y*;*u*(*t*) = *b*(*t*, *xs*,*y*;*u*(*t*), *u*(*t*))*dt* + *σ*(*t*, *xs*,*y*;*u*(*t*), *u*(*t*))*dW*(*t*)

Here *<sup>b</sup>* : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>→</sup> **<sup>R</sup>***n*, *<sup>σ</sup>* : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>→</sup> **<sup>R</sup>***n*×*d*, *<sup>c</sup>* : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>×</sup> **<sup>E</sup>** <sup>→</sup> **<sup>R</sup>***<sup>n</sup>* are

given *<sup>u</sup>*(·) ∈ U[*s*, *<sup>T</sup>*] and *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>***n*, a **<sup>R</sup>***n*-valued process *<sup>x</sup>s*,*y*;*u*(·) is called a solution of (4) if it is

*<sup>t</sup>* -adapted RCLL (i.e., right-continuous with left-hand limits) process such that (4) holds. It is called the *state trajectory* corresponding to the control *u*(·) ∈ U[*s*, *T*] with initial state *y*. We refer to such *<sup>u</sup>*(·) ∈ U[*s*, *<sup>T</sup>*] as an *admissible control* and (*xs*,*y*;*u*(·), *<sup>u</sup>*(·)) as an *admissible pair*.

� � *σ* �

*<sup>c</sup>*(*t*, *<sup>x</sup>s*,*y*;*u*(*t*−), *<sup>u</sup>*(*t*),*e*)*N*˜ (*dedt*), *<sup>t</sup>* <sup>∈</sup> (*s*, *<sup>T</sup>*],

*<sup>T</sup>*-measurable random variables *ξ*; **E**|*ξ*|

*<sup>t</sup>* -predictable vector processes *ψ*(*t*,*e*) defined on

<sup>2</sup>*π*(*de*)*dt* < ∞

� *T s*



� *T s*

> � .

> > �

*<sup>t</sup>* -adapted processes *ϕ*(*t*); **E**

*<sup>t</sup>* -predictable processes *φ*(*t*); **E**


*f*(*t*, *xs*,*y*;*u*(*t*), *u*(*t*))*dt* + *h*(*xs*,*y*;*u*(*T*))

<sup>L</sup><sup>2</sup> :<sup>=</sup> �

**<sup>E</sup>** |*k*(*e*)|

*<sup>t</sup>* }*s*≤*t*≤*<sup>T</sup>* is generated as the following

*W*(*r*) − *W*(*s*);*s* ≤ *r* ≤ *t*

� *<sup>σ</sup>*<sup>2</sup> denotes the *<sup>σ</sup>*-field generated by *<sup>σ</sup>*<sup>1</sup> ∪ *<sup>σ</sup>*2. In

<sup>2</sup>*π*(*de*) < ∞.

� � <sup>N</sup> ,

*<sup>t</sup>* -predictable processes. For

<sup>2</sup> < ∞ � ,

<sup>2</sup>*dt* < ∞

� ,

> � ,

<sup>2</sup>*dt* < ∞

, (5)

(4)

square integrable functions *<sup>k</sup>*(·) : **<sup>E</sup>** <sup>→</sup> **<sup>R</sup>***<sup>n</sup>* such that ||*k*(·)||<sup>2</sup>

*π*(*de*)*dt* for the compensated Poisson martingale measure.

*N*(**A** × (*s*,*r*]);*s* ≤ *r* ≤ *t*, **A** ∈ B(**E**)

+ � **E**

For a given *<sup>s</sup>* <sup>∈</sup> [0, *<sup>T</sup>*), we denote by <sup>U</sup>[*s*, *<sup>T</sup>*] the set of **<sup>U</sup>**-valued <sup>F</sup>*<sup>s</sup>*

*xs*,*y*;*u*(*s*) = *y*.

For any *s* ∈ [0, *T*), we introduce the following notations.

**<sup>R</sup>***n*-valued <sup>F</sup>*<sup>s</sup>*

**<sup>R</sup>***n*-valued <sup>F</sup>*<sup>s</sup>*

**<sup>R</sup>***n*-valued <sup>F</sup>*<sup>s</sup>*

**<sup>R</sup>***n*-valued <sup>F</sup>*<sup>s</sup>*

Ω × [0, *T*] × **E**; **E**

� � *<sup>T</sup> s*

� *T s* � **E**

�

�

�

�

*J*(*s*, *y*; *u*(·)) = **E**

We consider the following *cost functional*

*t* .

For a given *<sup>s</sup>* <sup>∈</sup> [0, *<sup>T</sup>*), we suppose the filtration {F*<sup>s</sup>*

where N contains all **P**-null sets in F and *σ*<sup>1</sup>

particular, if *<sup>s</sup>* <sup>=</sup> 0 we write <sup>F</sup>*<sup>t</sup>* ≡ F*<sup>s</sup>*

⎧ ⎪⎪⎪⎨

⎪⎪⎪⎩

*<sup>T</sup>*; **<sup>R</sup>***n*) :<sup>=</sup>

<sup>F</sup> ([*s*, *<sup>T</sup>*]; **<sup>R</sup>***n*) :<sup>=</sup>

<sup>F</sup>,*p*([*s*, *<sup>T</sup>*]; **<sup>R</sup>***n*) :<sup>=</sup>

*<sup>p</sup>* ([*s*, *<sup>T</sup>*]; **<sup>R</sup>***n*) :=

F*s <sup>t</sup>* := *σ* �

given functions.

*<sup>L</sup>*2(Ω, <sup>F</sup>*<sup>s</sup>*

*L*2

*F*2

*L*2

*process*

an <sup>F</sup>*<sup>s</sup>*

**Lemma 2.1** *Let* **(H1)** *hold. For k* = 2, 4*, there exists C* > 0 *such that for any* 0 ≤ *s*,*s*ˆ ≤ *T*, *y*, *y*ˆ ∈ **<sup>R</sup>***n*, *<sup>u</sup>*(·) ∈ U[*s*, *<sup>T</sup>*]*,*

$$\left|\mathbb{E}\left|\mathbf{x}^{\mathbf{s},\mathbf{y},\boldsymbol{\mu}}(t)\right|\right|^k \leq \mathbb{C}(1+|\boldsymbol{y}|^k), \quad t \in [\boldsymbol{s},T],\tag{7}$$

$$\mathbb{E}\left|\mathbf{x}^{s,y,\mu}(t) - y\right|^k \le \mathbb{C}(\mathbf{1} + |y|^k)(T - s)^{\frac{k}{2}}, \quad t \in [s, T],\tag{8}$$

$$\mathbb{E}\left[\sup\_{s\le t\le T}|\mathbf{x}^{s,y,\mu}(t)-\mathbf{y}|\right]^k\le\mathbb{C}(1+|y|^k)(T-s)^{\frac{k}{2}},\tag{9}$$

$$\mathbb{E}\left|\mathbf{x}^{s,y,\mu}(t) - \mathbf{x}^{s,\boldsymbol{\theta},\mu}(t)\right|^k \le \mathbb{C}\left|y - \boldsymbol{\mathcal{Y}}\right|^k, \quad t \in \left[\mathbf{s}, T\right],\tag{10}$$

$$\mathbb{E}\left|\mathbf{x}^{\mathbf{s},\boldsymbol{y},\boldsymbol{\mu}}(t) - \mathbf{x}^{\boldsymbol{\S},\boldsymbol{y},\boldsymbol{\mu}}(t)\right|^{k} \leq \mathbb{C}(1+|\boldsymbol{y}|) \left|s - \boldsymbol{\mathbb{s}}\right|^{\frac{k}{2}}, \quad t \in [\boldsymbol{s} \vee \boldsymbol{\S}, T]. \tag{11}$$

Estimates of the moments for SDEPs are proved in Lemma 3.1 of [14] for *k* ∈ [0, 2]. In fact, under assumption **(H1)** we can easily extend his result to the case *k* = 2, 4 by virtue of Buckholder-Davis-Gundy's inquality. We leave the detail of the proof to the interested reader.

We give some basic continuity properties of the value function *V*. The proof is similar to Proposition 2.2, Chapter 2 of [17]. We omit the detail.

**Proposition 2.1** *Let* **(H1)***,* **(H2)** *hold. Then there exist increasing, continuous functions ω*¯ <sup>1</sup> : [0, ∞) → [0, ∞)*, ω*¯ <sup>2</sup> : [0, ∞) × [0, ∞) → [0, ∞) *which satisfies ω*¯ <sup>2</sup>(*r*, 0) = 0, ∀*r* ≥ 0*, such that*

$$|V(s,y)| \le \bar{\omega}\_1(|y|), \quad \forall (s,y) \in [0,T) \times \mathbf{R}^n. \tag{12}$$

⎧



<sup>2</sup>), <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, *<sup>T</sup>*], *<sup>x</sup>*, *<sup>x</sup>*<sup>ˆ</sup> <sup>∈</sup> **<sup>R</sup>***n*, *<sup>u</sup>*, *<sup>u</sup>*<sup>ˆ</sup> <sup>∈</sup> **<sup>U</sup>**.

(*t*), *u*¯(*t*)),

(*t*−), *u*¯(*t*), ·),

*s*,*y*;*u*¯

�*γ*(*e*), *<sup>c</sup>*(*t*, *<sup>x</sup>*, *<sup>u</sup>*,*e*)�*π*(*de*). (17)

(·), *u*¯(·)), we introduce the following *first-order* and

(*t*), *<sup>u</sup>*¯(*t*), *<sup>p</sup>*(*t*), *<sup>q</sup>*(*t*), *<sup>γ</sup>*(*t*, ·))�

(*t*−), *u*, ·) − *c*¯(*t*, ·),

Stochastic Control for Jump Diff usions 125

(18)

(19)

(20)

*dt*

*<sup>p</sup>* ([*s*, *<sup>T</sup>*]; **<sup>R</sup>***n*),

*<sup>p</sup>* ([*s*, *<sup>T</sup>*]; <sup>S</sup>*n*).

*s*,*y*;*u*¯

*s*,*y*;*u*¯

(*t*), *u*¯(*t*), *p*(*t*), *q*(*t*), *γ*(*t*, ·))*dt* − *q*(*t*)*dW*(*t*)

*bx*(*t*) + *σ*¯*x*(*t*)�*P*(*t*)*σ*¯*x*(*t*) + *σ*¯*x*(*t*)�*Q*(*t*) + *Q*(*t*)*σ*¯*x*(*t*)

<sup>F</sup>,*p*([*s*, *<sup>T</sup>*]; **<sup>R</sup>***n*×*d*) <sup>×</sup> *<sup>F</sup>*<sup>2</sup>

�*d* <sup>×</sup> *<sup>F</sup>*<sup>2</sup>

<sup>F</sup>,*p*([*s*, *<sup>T</sup>*]; <sup>S</sup>*n*)

*c*¯*x*(*t*,*e*)�*P*(*t*)*c*¯*x*(*t*,*e*) + *c*¯*x*(*t*,*e*)�*R*(*t*,*e*)*c*¯*x*(*t*,*e*) + *c*¯*x*(*t*,*e*)�*R*(*t*,*e*)

*s*,*y*;*u*¯

*<sup>R</sup>*(*t*,*e*)*N*˜ (*dedt*), *<sup>t</sup>* <sup>∈</sup> [*s*, *<sup>T</sup>*],

*L*2

Under **(H1)**∼**(H3)**, by Lemma 2.4 of [16], we know that BSDEPs (18) and (19) admit unique



and similar notations used for all their derivatives, for all *t* ∈ [0, *T*], *u* ∈ **U**.

+ � **E**

*s*,*y*;*u*¯

*s*,*y*;*u*¯ (*T*)),

�

adapted solutions (*p*(·), *q*(·), *γ*(·, ·)) and (*P*(·), *Q*(·), *R*(·, ·)) satisfying

� **E**

(*t*), *u*¯(*t*)), *σ*¯(*t*) := *σ*(*t*, *x*¯

(*t*), *u*¯(*t*)), *c*¯(*t*, ·) := *c*(*t*, *x*¯

(*t*), *u*) − *σ*¯(*t*), Δ*c*(*t*, ·; *u*) := *c*(*t*, *x*¯

We define the *Hamiltonian function H* : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>R</sup>***n*×*<sup>d</sup>* × L2(**E**, <sup>B</sup>(**E**), *<sup>π</sup>*; **<sup>R</sup>***n*) <sup>→</sup> **<sup>R</sup>**:

*H*(*t*, *x*, *u*, *p*, *q*, *γ*(·)) := �*p*, *b*(*t*, *x*, *u*)� + tr{*q*�*σ*(*t*, *x*, *u*)} − *f*(*t*, *x*, *u*)

*<sup>γ</sup>*(*t*,*e*)*N*˜ (*dedt*), *<sup>t</sup>* <sup>∈</sup> [*s*, *<sup>T</sup>*],

*π*(*de*) + *Hxx*(*t*, *x*¯

<sup>F</sup> ([*s*, *<sup>T</sup>*]; **<sup>R</sup>***n*) <sup>×</sup> *<sup>L</sup>*<sup>2</sup>

<sup>F</sup> ([*s*, *<sup>T</sup>*]; <sup>S</sup>*n*) <sup>×</sup> �


For simplicity, we introduce the following notations.

*s*,*y*;*u*¯

*s*,*y*;*u*¯

*s*,*y*;*u*¯

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

¯

¯

*b*(*t*) := *b*(*t*, *x*¯

*f*(*t*) := *f*(*t*, *x*¯

Associated with an optimal pair (*x*¯*s*,*y*;*u*¯

⎧ ⎪⎪⎪⎨

⎪⎪⎪⎩

*P*(*T*) = −*hxx*(*x*¯

⎧ ⎨ ⎩ ¯

+ � **E** �

<sup>−</sup>*dP*(*t*) = �

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

*second-order adjoint equations*, respectively:

−*dp*(*t*) = *Hx*(*t*, *x*¯

− � **E**

*p*(*T*) = −*hx*(*x*¯

*bx*(*t*)�*P*(*t*) + *P*(*t*)¯

+ *R*(*t*,*e*)*c*¯*x*(*t*,*e*)

*s*,*y*;*u*¯ (*T*)),

− *Q*(*t*)*dW*(*t*) −

(*p*(·), *<sup>q</sup>*(·), *<sup>γ</sup>*(·, ·)) <sup>∈</sup> *<sup>L</sup>*<sup>2</sup>

(*P*(·), *<sup>Q</sup>*(·), *<sup>R</sup>*(·, ·)) <sup>∈</sup> *<sup>L</sup>*<sup>2</sup>

Δ*σ*(*t*; *u*) := *σ*(*t*, *x*¯

$$|V(\mathbf{s},\boldsymbol{y}) - V(\boldsymbol{\mathfrak{s}},\boldsymbol{\mathfrak{y}})| \le \bar{\omega}\_2 \left( |\boldsymbol{y}| \vee |\boldsymbol{\mathfrak{y}}| \, \_{\prime} |\boldsymbol{y} - \boldsymbol{\mathfrak{y}}| + |\boldsymbol{\mathfrak{s}} - \boldsymbol{\mathfrak{s}}|^{\frac{1}{2}} \right), \quad \forall \mathbf{s}, \boldsymbol{\mathfrak{s}} \in \left[ 0, T \right), \boldsymbol{y}, \boldsymbol{\mathfrak{y}} \in \mathbf{R}^n. \tag{13}$$

The following result is a version of *Bellman's principle of optimality* for jump diffusions.

**Theorem 2.1** *Suppose that* **(H1), (H2)** *hold. Then for any* (*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n,*

$$V(\mathbf{s}, y) = \inf\_{\mathbf{u}(\cdot) \in \mathcal{U}[\mathbf{s}, T]} \mathbb{E} \left[ \int\_{s}^{\mathbb{S}} f(t, \mathbf{x}^{\mathbf{s}, y; \boldsymbol{\mu}}(t), \mathbf{u}(t)) dt + V(\mathbf{s}, \mathbf{x}^{\mathbf{s}, y; \boldsymbol{\mu}}(\boldsymbol{\xi})) \right], \quad \forall 0 \le s \le \mathbb{S} \le T. \tag{14}$$

The proof is similar to Theorem 3.3, Chapter 4 of [18] or Proposition 3.2 of [14]. We omit it here.

The following result is to get the *generalized HJB equation* and its proof is similar to Proposition 3.4, Chapter 4 of [18].

**Theorem 2.2** *Suppose that* **(H1), (H2)** *hold and the value function V* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*. Then V is a solution of the following generalized HJB equation which is a second-order PIDE:*

$$\begin{cases} -V\_t(t, \mathbf{x}) + \sup\_{\mathbf{u} \in \mathbf{U}} G(t, \mathbf{x}, \mathbf{u}, -V(t, \mathbf{x}), -V\_\mathbf{x}(t, \mathbf{x}), -V\_\mathbf{x}(t, \mathbf{x})) = 0, \quad (t, \mathbf{x}) \in [0, T) \times \mathbb{R}^n, \\\\ V(T, \mathbf{x}) = h(\mathbf{x}), \qquad \mathbf{x} \in \mathbb{R}^n, \end{cases} \tag{15}$$

*where, associated with a* <sup>Ψ</sup> <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*, the generalized Hamiltonian function G* : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>×</sup> **<sup>R</sup>** <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* × S*<sup>n</sup>* <sup>→</sup> **<sup>R</sup>** is defined by

$$\begin{split} &G(t,\mathbf{x},\boldsymbol{u},\boldsymbol{\Psi}(t,\mathbf{x}),\boldsymbol{\Psi}\_{\mathbf{x}}(t,\mathbf{x}),\boldsymbol{\Psi}\_{\mathbf{xx}}(t,\mathbf{x})) \\ &:= \left\langle \boldsymbol{\Psi}\_{\mathbf{x}}(t,\mathbf{x}),\boldsymbol{b}(t,\mathbf{x},\boldsymbol{u}) \right\rangle + \frac{1}{2} \text{tr} \Big\{ \boldsymbol{\Psi}\_{\mathbf{xx}}(t,\mathbf{x}) \boldsymbol{\sigma}(t,\mathbf{x},\boldsymbol{u}) \boldsymbol{\sigma}(t,\mathbf{x},\boldsymbol{u})^{\top} \Big\} \\ & \qquad \quad - f(t,\mathbf{x},\boldsymbol{u}) - \int\_{\mathbf{E}} \Big[ \boldsymbol{\Psi}(t,\mathbf{x}+\mathbf{c}(t,\mathbf{x},\boldsymbol{u},\boldsymbol{e})) - \boldsymbol{\Psi}(t,\mathbf{x}) + \langle \boldsymbol{\Psi}\_{\mathbf{x}}(t,\mathbf{x}),\mathbf{c}(t,\mathbf{x},\boldsymbol{u},\boldsymbol{e}) \rangle \Big] \pi(d\mathbf{e}). \end{split} \tag{16}$$

In the following, we discuss the stochastic MP and need the following hypothesis.

**(H3)** *b*, *σ*, *c*, *f* , *h* are twice continuously differentiable in *x*, and *bx*, *bxx*, *σx*, *σxx*, *fxx*, *hxx*, ||*cx*(·)||L<sup>2</sup> , ||*cxx*(·)||L<sup>2</sup> are bounded. There exists a constant *<sup>C</sup>* <sup>&</sup>gt; 0 and a modulus of continuity *ω*¯ : [0, ∞) → [0, ∞) such that

⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ |*bx*(*t*, *x*, *u*) − *bx*(*t*, *x*ˆ, *u*ˆ)| + |*σx*(*t*, *x*, *u*) − *σx*(*t*, *x*ˆ, *u*ˆ)| ≤ *C*|*x* − *x*ˆ| + *ω*¯(|*u* − *u*ˆ|), ||*cx*(*t*, *<sup>x</sup>*, *<sup>u</sup>*, ·) <sup>−</sup> *cx*(*t*, *<sup>x</sup>*ˆ, *<sup>u</sup>*ˆ, ·)||L<sup>2</sup> <sup>≤</sup> *<sup>C</sup>*|*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*ˆ<sup>|</sup> <sup>+</sup> *<sup>ω</sup>*¯(|*<sup>u</sup>* <sup>−</sup> *<sup>u</sup>*ˆ|), |*bxx*(*t*, *x*, *u*) − *bxx*(*t*, *x*ˆ, *u*ˆ)| + |*σxx*(*t*, *x*, *u*) − *σxx*(*t*, *x*ˆ, *u*ˆ)| ≤ *ω*¯(|*x* − *x*ˆ| + |*u* − *u*ˆ|), ||*cxx*(*t*, *<sup>x</sup>*, *<sup>u</sup>*, ·) <sup>−</sup> *cxx*(*t*, *<sup>x</sup>*ˆ, *<sup>u</sup>*ˆ, ·)||L<sup>2</sup> <sup>≤</sup> *<sup>ω</sup>*¯(|*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*ˆ<sup>|</sup> <sup>+</sup> <sup>|</sup>*<sup>u</sup>* <sup>−</sup> *<sup>u</sup>*ˆ|), | *fx*(*t*, *x*, *u*)| + |*hx*(*x*)| ≤ *C*(1 + |*x*|), | *f*(*t*, *x*, *u*)| + |*h*(*x*)| ≤ *C*(1 + |*x*| <sup>2</sup>), <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, *<sup>T</sup>*], *<sup>x</sup>*, *<sup>x</sup>*<sup>ˆ</sup> <sup>∈</sup> **<sup>R</sup>***n*, *<sup>u</sup>*, *<sup>u</sup>*<sup>ˆ</sup> <sup>∈</sup> **<sup>U</sup>**.

For simplicity, we introduce the following notations.

6 Will-be-set-by-IN-TECH

Estimates of the moments for SDEPs are proved in Lemma 3.1 of [14] for *k* ∈ [0, 2]. In fact, under assumption **(H1)** we can easily extend his result to the case *k* = 2, 4 by virtue of Buckholder-Davis-Gundy's inquality. We leave the detail of the proof to the interested reader. We give some basic continuity properties of the value function *V*. The proof is similar to

**Proposition 2.1** *Let* **(H1)***,* **(H2)** *hold. Then there exist increasing, continuous functions ω*¯ <sup>1</sup> : [0, ∞) → [0, ∞)*, ω*¯ <sup>2</sup> : [0, ∞) × [0, ∞) → [0, ∞) *which satisfies ω*¯ <sup>2</sup>(*r*, 0) = 0, ∀*r* ≥ 0*, such that*


The following result is a version of *Bellman's principle of optimality* for jump diffusions.

*f*(*t*, *xs*,*y*;*u*(*t*), *u*(*t*))*dt* + *V*(*s*ˆ, *xs*,*y*;*u*(*s*ˆ))

The proof is similar to Theorem 3.3, Chapter 4 of [18] or Proposition 3.2 of [14]. We omit it

The following result is to get the *generalized HJB equation* and its proof is similar to Proposition

**Theorem 2.2** *Suppose that* **(H1), (H2)** *hold and the value function V* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*. Then V*

*where, associated with a* <sup>Ψ</sup> <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*, the generalized Hamiltonian function G* : [0, *<sup>T</sup>*] <sup>×</sup>

<sup>Ψ</sup>*xx*(*t*, *<sup>x</sup>*)*σ*(*t*, *<sup>x</sup>*, *<sup>u</sup>*)*σ*(*t*, *<sup>x</sup>*, *<sup>u</sup>*)��

In the following, we discuss the stochastic MP and need the following hypothesis.

**(H3)** *b*, *σ*, *c*, *f* , *h* are twice continuously differentiable in *x*, and *bx*, *bxx*, *σx*, *σxx*, *fxx*, *hxx*,


Ψ(*t*, *x* + *c*(*t*, *x*, *u*,*e*)) − Ψ(*t*, *x*) + �Ψ*x*(*t*, *x*), *c*(*t*, *x*, *u*,*e*)�

*<sup>G</sup>*(*t*, *<sup>x</sup>*, *<sup>u</sup>*, <sup>−</sup>*V*(*t*, *<sup>x</sup>*), <sup>−</sup>*Vx*(*t*, *<sup>x</sup>*), <sup>−</sup>*Vxx*(*t*, *<sup>x</sup>*)) = 0, (*t*, *<sup>x</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n*,

**Theorem 2.1** *Suppose that* **(H1), (H2)** *hold. Then for any* (*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n,*

*is a solution of the following generalized HJB equation which is a second-order PIDE:*

<sup>|</sup>*V*(*s*, *<sup>y</sup>*)| ≤ *<sup>ω</sup>*¯ <sup>1</sup>(|*y*|), <sup>∀</sup>(*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n*, (12)

�

, <sup>∀</sup>*s*,*s*<sup>ˆ</sup> <sup>∈</sup> [0, *<sup>T</sup>*), *<sup>y</sup>*, *<sup>y</sup>*<sup>ˆ</sup> <sup>∈</sup> **<sup>R</sup>***n*. (13)

, ∀0 ≤ *s* ≤ *s*ˆ ≤ *T*. (14)

� *π*(*de*). (15)

(16)

1 2 | �

Proposition 2.2, Chapter 2 of [17]. We omit the detail.

�

**E** � � *<sup>s</sup>*<sup>ˆ</sup> *s*


*<sup>V</sup>*(*s*, *<sup>y</sup>*) = inf *<sup>u</sup>*(·)∈U[*s*,*T*]

3.4, Chapter 4 of [18].

− *Vt*(*t*, *x*) + sup

*u*∈**U**

*<sup>V</sup>*(*T*, *<sup>x</sup>*) = *<sup>h</sup>*(*x*), *<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n*,

:= �Ψ*x*(*t*, *x*), *b*(*t*, *x*, *u*)� +

− *f*(*t*, *x*, *u*) −

*ω*¯ : [0, ∞) → [0, ∞) such that

**<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>×</sup> **<sup>R</sup>** <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* × S*<sup>n</sup>* <sup>→</sup> **<sup>R</sup>** is defined by

*G*(*t*, *x*, *u*, Ψ(*t*, *x*), Ψ*x*(*t*, *x*), Ψ*xx*(*t*, *x*))

� **E** � 1 2 tr �

here.

⎧ ⎪⎨

⎪⎩

$$\begin{aligned} \bar{b}(t) &:= b(t, \bar{\mathfrak{x}}^{\mathbb{S}, \mathbb{Y}; \mathbb{R}}(t), \bar{u}(t)), \quad \bar{\sigma}(t) := \sigma(t, \bar{\mathfrak{x}}^{\mathbb{S}, \mathbb{Y}; \mathbb{R}}(t), \bar{u}(t)), \\ \bar{f}(t) &:= f(t, \bar{\mathfrak{x}}^{\mathbb{S}, \mathbb{Y}; \mathbb{R}}(t), \bar{u}(t)), \quad \bar{\varepsilon}(t, \cdot) := c(t, \bar{\mathfrak{x}}^{\mathbb{S}, \mathbb{Y}; \mathbb{R}}(t-), \bar{u}(t), \cdot), \\ \Delta \sigma(t; \boldsymbol{u}) &:= \sigma(t, \bar{\mathfrak{x}}^{\mathbb{S}, \mathbb{Y}; \mathbb{R}}(t), \boldsymbol{u}) - \bar{\sigma}(t), \quad \Delta c(t, \cdot; \boldsymbol{u}) := c(t, \bar{\mathfrak{x}}^{\mathbb{S}, \mathbb{Y}; \mathbb{R}}(t-), \boldsymbol{u}\_{\prime} \cdot) - \bar{\varepsilon}(t, \cdot), \end{aligned}$$

and similar notations used for all their derivatives, for all *t* ∈ [0, *T*], *u* ∈ **U**. We define the *Hamiltonian function H* : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>R</sup>***n*×*<sup>d</sup>* × L2(**E**, <sup>B</sup>(**E**), *<sup>π</sup>*; **<sup>R</sup>***n*) <sup>→</sup> **<sup>R</sup>**:

$$\begin{split} H(t, \mathbf{x}, \boldsymbol{\mu}, \boldsymbol{p}, \boldsymbol{q}, \boldsymbol{\gamma}(\cdot)) &:= \langle p, b(t, \mathbf{x}, \boldsymbol{\mu}) \rangle + \text{tr} \{ q^{\top} \boldsymbol{\sigma}(t, \mathbf{x}, \boldsymbol{\mu}) \} - f(t, \mathbf{x}, \boldsymbol{\mu}) \\ &\quad + \int\_{\mathbf{E}} \langle \boldsymbol{\gamma}(\boldsymbol{e}), c(t, \mathbf{x}, \boldsymbol{\mu}, \boldsymbol{e}) \rangle \boldsymbol{\pi}(d\mathbf{e}). \end{split} \tag{17}$$

Associated with an optimal pair (*x*¯*s*,*y*;*u*¯ (·), *u*¯(·)), we introduce the following *first-order* and *second-order adjoint equations*, respectively:

$$\begin{cases} -dp(t) = H\_{\mathcal{X}}(t, \bar{\mathfrak{x}}^{\mathcal{S}; \mathcal{U}}(t), \bar{u}(t), p(t), q(t), \gamma(t, \cdot))dt - q(t)dW(t) \\ \qquad - \int\_{\mathbb{E}} \gamma(t, e)\mathcal{N}(dedt), \; t \in [\mathcal{s}, T], \\\ p(T) = -h\_{\mathcal{X}}(\bar{\mathfrak{x}}^{\mathcal{S}; \mathcal{U}}(T)), \end{cases} \tag{18}$$

$$\begin{cases} -d P(t) = \left\{ \bar{b}\_{\mathcal{X}}(t)^{\top} P(t) + P(t) \bar{b}\_{\mathcal{X}}(t) + \bar{\sigma}\_{\mathcal{X}}(t)^{\top} P(t) \bar{\sigma}\_{\mathcal{X}}(t) + \bar{\sigma}\_{\mathcal{X}}(t)^{\top} Q(t) + Q(t) \bar{\sigma}\_{\mathcal{X}}(t) \right\} \\ \qquad + \int\_{\mathcal{E}} \left[ \bar{c}\_{\mathcal{X}}(t, e)^{\top} P(t) \bar{c}\_{\mathcal{X}}(t, e) + \bar{c}\_{\mathcal{X}}(t, e)^{\top} R(t, e) \bar{c}\_{\mathcal{X}}(t, e) + \bar{c}\_{\mathcal{X}}(t, e)^{\top} R(t, e) \right. \\ \left. \quad + R(t, e) \bar{c}\_{\mathcal{X}}(t, e) \right] \pi(de) + H\_{\text{xx}}(t, \bar{\mathfrak{x}}^{\sharp, \mathcal{Y}; \mathcal{B}}(t), \bar{\mathfrak{u}}(t), p(t), q(t), \gamma(t, \cdot)) \Bigg\} dt \\ \qquad - Q(t) dW(t) - \int\_{\mathcal{E}} R(t, e) \bar{\mathcal{N}}(de dt), \qquad t \in [s, T], \\ P(T) = -h\_{\text{xx}}(\bar{\mathfrak{x}}^{\sharp, \mathcal{Y}; \mathcal{B}}(T)), \end{cases} \tag{19}$$

Under **(H1)**∼**(H3)**, by Lemma 2.4 of [16], we know that BSDEPs (18) and (19) admit unique adapted solutions (*p*(·), *q*(·), *γ*(·, ·)) and (*P*(·), *Q*(·), *R*(·, ·)) satisfying

$$\begin{cases} \begin{array}{l} \left(p(\cdot),q(\cdot),\gamma(\cdot,\cdot)\right) \in L^{2}\_{\mathcal{F}}([\sf s,T];\mathbf{R}^{\sf n}) \times L^{2}\_{\mathcal{F},p}([\sf s,T];\mathbf{R}^{\sf n \times d}) \times F^{2}\_{p}([\sf s,T];\mathbf{R}^{\sf n}),\\ \left(P(\cdot),Q(\cdot),\mathcal{R}(\cdot,\cdot)\right) \in L^{2}\_{\mathcal{F}}([\sf s,T];\mathbf{S}^{\sf n}) \times \left(L^{2}\_{\mathcal{F},p}([\sf s,T];\mathbf{S}^{\sf n})\right)^{d} \times F^{2}\_{p}([\sf s,T];\mathbf{S}^{\sf n}). \end{array} \end{cases} \tag{20}$$

Note that *<sup>p</sup>*(·) and *<sup>P</sup>*(·) are RCLL processes. Associated with an optimal pair (*x*¯*s*,*y*;*u*¯ (·), *u*¯(·)) and its corresponding adjoint processes (*p*(·), *q*(·), *γ*(·, ·)) and (*P*(·), *Q*(·), *R*(·, ·)) satisfying (20), we define an <sup>H</sup>-function <sup>H</sup> : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>→</sup> **<sup>R</sup>** as

**3. Relationship between Stochastic MP and DPP**

**3.1. Preliminary results: viscosity solutions and semijets**

also given to illustrate our results.

HJB equation (15) in our jump diffusion setting.

− *ψt*(*t*, *x*) + sup

− *ψt*(*t*, *x*) + sup

+ 1 2

+ 1 2

and the *right parabolic subjet* of *v* at (ˆ*t*, *x*ˆ) is the set

*u*∈**U**

*u*∈**U**

(*q*, *<sup>p</sup>*, *<sup>P</sup>*) <sup>∈</sup> **<sup>R</sup>** <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* × S*<sup>n</sup>*

(*q*, *<sup>p</sup>*, *<sup>P</sup>*) <sup>∈</sup> **<sup>R</sup>** <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* × S*<sup>n</sup>*

*global maximum at* (*t*, *<sup>x</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n, then*

(*t*, *<sup>x</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n, then*

is the set triple

P1,2,<sup>+</sup>

*<sup>t</sup>*+,*<sup>x</sup> <sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ) :=

<sup>P</sup>1,2,<sup>−</sup> *<sup>t</sup>*+,*<sup>x</sup> <sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ) :<sup>=</sup>

*called a viscosity solution of (15).*

In this section, we will establish the relationship between stochastic MP and DPP in the language of viscosity solutions. That is to say, we will consider the viscosity solutions of the generalized HJB equation (15). In our jump diffusion setting, we need use the viscosity solution theory for second-order PIDEs. For convenience, we refer to [1], [2], [14], [3], [11], [6] for a deep investigation of PIDEs in the framework of viscosity solutions. In Subsection 3.1, we first present some preliminary results concerning viscosity solutions and semijets. Then we give the relationship between stochastic MP and DPP in Subsection 3.2. Special cases on the assumption that the value function is smooth are given as corollaries. Some examples are

To make the chapter self-contained, we present the definition of viscosity solutions and semijets, which is frequently seen in the literature and we state it adapting to the generalized

**Definition 3.1** *(i) A function v* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *is called a viscosity subsolution of (15) if <sup>v</sup>*(*T*, *<sup>x</sup>*) <sup>≤</sup> *<sup>h</sup>*(*x*), <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n*, *and for any test function <sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*, whenever v* <sup>−</sup> *<sup>ψ</sup> attains a*

*(ii) A function v* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *is called a viscosity supersolution of (15) if v*(*T*, *<sup>x</sup>*) <sup>≥</sup> *<sup>h</sup>*(*x*), <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n, and for any test function <sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*, whenever v* <sup>−</sup> *<sup>ψ</sup> attains a global minimum at*

*(iii) If v* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *is both a viscosity subsolution and viscosity supersolution of (15), then it is*

In order to give the existence and uniqueness result for viscosity solution of the generalized HJB equation (15), it is convenient to give an intrinsic characterization of viscosity solutions. Let us recall the right parabolic super-subjets of a continuous function on [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* (see [18] or [3]). For *<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) and (ˆ*t*, *<sup>x</sup>*ˆ) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n*, the *right parabolic superjet* of *<sup>v</sup>* at (ˆ*t*, *<sup>x</sup>*ˆ)

(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*ˆ)�*P*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*ˆ) + *<sup>o</sup>*(|*<sup>t</sup>* <sup>−</sup> <sup>ˆ</sup>*t*<sup>|</sup> <sup>+</sup> <sup>|</sup>*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*ˆ<sup>|</sup>

(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*ˆ)�*P*(*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*ˆ) + *<sup>o</sup>*(|*<sup>t</sup>* <sup>−</sup> <sup>ˆ</sup>*t*<sup>|</sup> <sup>+</sup> <sup>|</sup>*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*ˆ<sup>|</sup>

*G*(*t*, *x*, *u*, −*ψ*(*t*, *x*), −*ψx*(*t*, *x*), −*ψxx*(*t*, *x*)) ≤ 0. (24)

*G*(*t*, *x*, *u*, −*ψ*(*t*, *x*), −*ψx*(*t*, *x*), −*ψxx*(*t*, *x*)) ≥ 0. (25)

*<sup>v</sup>*(*t*, *<sup>x</sup>*) <sup>≤</sup> *<sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ) + *<sup>q</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>ˆ</sup>*t*) + �*p*, *<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*ˆ�

*<sup>v</sup>*(*t*, *<sup>x</sup>*) <sup>≥</sup> *<sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ) + *<sup>q</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>ˆ</sup>*t*) + �*p*, *<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*ˆ�

<sup>2</sup>), as *<sup>t</sup>* <sup>↓</sup> <sup>ˆ</sup>*t*, *<sup>x</sup>* <sup>→</sup> *<sup>x</sup>*<sup>ˆ</sup>

<sup>2</sup>), as *<sup>t</sup>* <sup>↓</sup> <sup>ˆ</sup>*t*, *<sup>x</sup>* <sup>→</sup> *<sup>x</sup>*<sup>ˆ</sup>

 ,

Stochastic Control for Jump Diff usions 127

 . (26)

(27)

<sup>H</sup>(*t*, *<sup>x</sup>*, *<sup>u</sup>*) :<sup>=</sup> *<sup>H</sup>*(*t*, *<sup>x</sup>*, *<sup>u</sup>*, *<sup>p</sup>*(*t*), *<sup>q</sup>*(*t*), *<sup>γ</sup>*(*t*, ·)) <sup>−</sup> <sup>1</sup> 2 tr *P*(*t*) *σ*¯(*t*)*σ*¯(*t*)� + **E** *c*¯(*t*,*e*)*c*¯(*t*,*e*)�*π*(*de*) + 1 2 tr *P*(*t*) Δ*σ*(*t*; *u*)Δ*σ*(*t*; *u*)� + **E** Δ*c*(*t*,*e*; *u*)Δ*c*(*t*,*e*; *u*)�*π*(*de*) − 1 2 tr **E** *R*(*t*,*e*) *<sup>c</sup>*¯(*t*,*e*)*c*¯(*t*,*e*)� <sup>−</sup> <sup>Δ</sup>*c*(*t*,*e*; *<sup>u</sup>*)Δ*c*(*t*,*e*; *<sup>u</sup>*)� *π*(*de*) ≡ �*p*(*t*), *b*(*t*, *x*, *u*)� + tr{*q*(*t*), *σ*(*t*, *x*, *u*)} + **E** �*γ*(*t*,*e*), *c*(*t*, *x*, *u*,*e*)�*π*(*de*) <sup>−</sup> *<sup>f</sup>*(*t*, *<sup>x</sup>*, *<sup>u</sup>*) <sup>−</sup> <sup>1</sup> 2 tr *P*(*t*) *σ*¯(*t*)*σ*¯(*t*)� + **E** *c*¯(*t*,*e*)*c*¯(*t*,*e*)�*π*(*de*) + 1 2 tr *P*(*t*) Δ*σ*(*t*; *u*)Δ*σ*(*t*; *u*)� + **E** Δ*c*(*t*,*e*; *u*)Δ*c*(*t*,*e*; *u*)�*π*(*de*) − 1 2 tr **E** *R*(*t*,*e*) *<sup>c</sup>*¯(*t*,*e*)*c*¯(*t*,*e*)� <sup>−</sup> <sup>Δ</sup>*c*(*t*,*e*; *<sup>u</sup>*)Δ*c*(*t*,*e*; *<sup>u</sup>*)� *π*(*de*) . (21)

The following result is the general stochastic MP for jump diffusions.

**Theorem 2.3** *Suppose that* **(H1)**∼**(H3)** *hold. Let* (*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> be fixed and* (*x*¯*s*,*y*;*u*¯ (·), *u*¯(·)) *be an optimal pair of our stochastic optimal control problem. Then there exist triples of processes* (*p*(·), *q*(·), *γ*(·, ·)) *and* (*P*(·), *Q*(·), *R*(·, ·)) *satisfying (20) and they are solutions of first-order and second-order adjoint equations (18) and (19), respectively, such that the following maximum condition holds:*

$$\begin{split} &H(t, \check{\mathbf{x}}^{\ $j\|t}(t), \check{u}(t), p(t), q(t), \gamma(t, \cdot)) - H(t, \check{\mathbf{x}}^{\$ j\|t}(t), u, p(t), q(t), \gamma(t, \cdot)) \\ &- \frac{1}{2} \text{tr} \Big\{ P(t) \Big[ \Delta \sigma(t; u) \Delta \sigma(t; u)^{\top} + \int\_{\mathbf{E}} \Delta \boldsymbol{\varepsilon}(t, \boldsymbol{\varepsilon}; u) \Delta \boldsymbol{\varepsilon}(t, \boldsymbol{\varepsilon}; u)^{\top} \pi(d\boldsymbol{\varepsilon}) \Big] \Big\} \\ &- \frac{1}{2} \text{tr} \Big\{ \int\_{\mathbf{E}} R(t, \boldsymbol{\varepsilon}) \Big[ \Delta \boldsymbol{\varepsilon}(t, \boldsymbol{\varepsilon}; u) \Delta \boldsymbol{\varepsilon}(t, \boldsymbol{\varepsilon}; u)^{\top} \Big] \pi(d\boldsymbol{\varepsilon}) \Big\} \geq 0, \quad \forall u \in \mathbf{U}, a.e.t \in [s\_{\boldsymbol{\prime}} T]\_{\boldsymbol{\prime}} \mathbf{P} \text{-a.s.}, \end{split} \tag{22}$$

*or equivalently,*

$$\mathcal{H}(t, \vec{\boldsymbol{x}}(t), \vec{\boldsymbol{u}}(t)) = \max\_{\boldsymbol{\mu} \in \mathbf{U}} \mathcal{H}(t, \vec{\boldsymbol{x}}(t), \boldsymbol{\mu}), \quad \text{a.e.} \boldsymbol{t} \in [\mathbf{s}\_{\prime} \, T]\_{\prime} \quad \text{P-a.s.} \tag{23}$$

*Proof* It is an immediate consequence of Theorem 2.1 of [16]. The equivalence of (22) and (23) is obvious.

**Remark 2.2** Note that the integrand with respect to the compensated martingale measure *N*˜ in the second-order adjoint equation enters into the above maximum condition, while the counterpart in the diffusion case does not! This marks one essential difference of the maximum principle between an optimally controlled diffusion (*continuous*) process and an optimally controlled jump (*discontinuous*) process.

## **3. Relationship between Stochastic MP and DPP**

8 Will-be-set-by-IN-TECH

and its corresponding adjoint processes (*p*(·), *q*(·), *γ*(·, ·)) and (*P*(·), *Q*(·), *R*(·, ·)) satisfying

*σ*¯(*t*)*σ*¯(*t*)� +

 **E**

�*γ*(*t*,*e*), *c*(*t*, *x*, *u*,*e*)�*π*(*de*)

Δ*c*(*t*,*e*; *u*)Δ*c*(*t*,*e*; *u*)�*π*(*de*)

*c*¯(*t*,*e*)*c*¯(*t*,*e*)�*π*(*de*)

Δ*c*(*t*,*e*; *u*)Δ*c*(*t*,*e*; *u*)�*π*(*de*)

(*t*), *u*, *p*(*t*), *q*(*t*), *γ*(*t*, ·))

≥ 0, ∀*u* ∈ **U**, *a*.*e*.*t* ∈ [*s*, *T*], **P**-*a*.*s*.,

H(*t*, *x*¯(*t*), *u*), *a*.*e*.*t* ∈ [*s*, *T*], **P***-a*.*s*. (23)

*c*¯(*t*,*e*)*c*¯(*t*,*e*)�*π*(*de*)

*π*(*de*) 

*π*(*de*) .

2 tr *P*(*t*) 

*σ*¯(*t*)*σ*¯(*t*)� +

 **E**

 **E**

*<sup>c</sup>*¯(*t*,*e*)*c*¯(*t*,*e*)� <sup>−</sup> <sup>Δ</sup>*c*(*t*,*e*; *<sup>u</sup>*)Δ*c*(*t*,*e*; *<sup>u</sup>*)�

 **E**

 **E**

*<sup>c</sup>*¯(*t*,*e*)*c*¯(*t*,*e*)� <sup>−</sup> <sup>Δ</sup>*c*(*t*,*e*; *<sup>u</sup>*)Δ*c*(*t*,*e*; *<sup>u</sup>*)�

Δ*σ*(*t*; *u*)Δ*σ*(*t*; *u*)� +

Δ*σ*(*t*; *u*)Δ*σ*(*t*; *u*)� +

 **E**

*u*∈**U**

**Theorem 2.3** *Suppose that* **(H1)**∼**(H3)** *hold. Let* (*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> be fixed and* (*x*¯*s*,*y*;*u*¯

*be an optimal pair of our stochastic optimal control problem. Then there exist triples of processes* (*p*(·), *q*(·), *γ*(·, ·)) *and* (*P*(·), *Q*(·), *R*(·, ·)) *satisfying (20) and they are solutions of first-order and second-order adjoint equations (18) and (19), respectively, such that the following maximum condition*

*s*,*y*;*u*¯

*π*(*de*) 

*Proof* It is an immediate consequence of Theorem 2.1 of [16]. The equivalence of (22) and

**Remark 2.2** Note that the integrand with respect to the compensated martingale measure *N*˜ in the second-order adjoint equation enters into the above maximum condition, while the counterpart in the diffusion case does not! This marks one essential difference of the maximum principle between an optimally controlled diffusion (*continuous*) process and an

Δ*c*(*t*,*e*; *u*)Δ*c*(*t*,*e*; *u*)�*π*(*de*)

The following result is the general stochastic MP for jump diffusions.

(*t*), *u*¯(*t*), *p*(*t*), *q*(*t*), *γ*(*t*, ·)) − *H*(*t*, *x*¯

<sup>Δ</sup>*c*(*t*,*e*; *<sup>u</sup>*)Δ*c*(*t*,*e*; *<sup>u</sup>*)�

Δ*σ*(*t*; *u*)Δ*σ*(*t*; *u*)� +

H(*t*, *x*¯(*t*), *u*¯(*t*)) = max

optimally controlled jump (*discontinuous*) process.

≡ �*p*(*t*), *b*(*t*, *x*, *u*)� + tr{*q*(*t*), *σ*(*t*, *x*, *u*)} +

2 tr *P*(*t*)  (·), *u*¯(·))

(21)

(22)

(·), *u*¯(·))

Note that *<sup>p</sup>*(·) and *<sup>P</sup>*(·) are RCLL processes. Associated with an optimal pair (*x*¯*s*,*y*;*u*¯

(20), we define an <sup>H</sup>-function <sup>H</sup> : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>→</sup> **<sup>R</sup>** as

<sup>H</sup>(*t*, *<sup>x</sup>*, *<sup>u</sup>*) :<sup>=</sup> *<sup>H</sup>*(*t*, *<sup>x</sup>*, *<sup>u</sup>*, *<sup>p</sup>*(*t*), *<sup>q</sup>*(*t*), *<sup>γ</sup>*(*t*, ·)) <sup>−</sup> <sup>1</sup>

<sup>−</sup> *<sup>f</sup>*(*t*, *<sup>x</sup>*, *<sup>u</sup>*) <sup>−</sup> <sup>1</sup>

+ 1 2 tr *P*(*t*) 

− 1 2 tr **E** *R*(*t*,*e*) 

+ 1 2 tr *P*(*t*) 

− 1 2 tr **E** *R*(*t*,*e*) 

*holds:*

*H*(*t*, *x*¯

− 1 2 tr *P*(*t*) 

− 1 2 tr **E** *R*(*t*,*e*) 

*or equivalently,*

(23) is obvious.

*s*,*y*;*u*¯

In this section, we will establish the relationship between stochastic MP and DPP in the language of viscosity solutions. That is to say, we will consider the viscosity solutions of the generalized HJB equation (15). In our jump diffusion setting, we need use the viscosity solution theory for second-order PIDEs. For convenience, we refer to [1], [2], [14], [3], [11], [6] for a deep investigation of PIDEs in the framework of viscosity solutions. In Subsection 3.1, we first present some preliminary results concerning viscosity solutions and semijets. Then we give the relationship between stochastic MP and DPP in Subsection 3.2. Special cases on the assumption that the value function is smooth are given as corollaries. Some examples are also given to illustrate our results.

#### **3.1. Preliminary results: viscosity solutions and semijets**

To make the chapter self-contained, we present the definition of viscosity solutions and semijets, which is frequently seen in the literature and we state it adapting to the generalized HJB equation (15) in our jump diffusion setting.

**Definition 3.1** *(i) A function v* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *is called a viscosity subsolution of (15) if <sup>v</sup>*(*T*, *<sup>x</sup>*) <sup>≤</sup> *<sup>h</sup>*(*x*), <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n*, *and for any test function <sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*, whenever v* <sup>−</sup> *<sup>ψ</sup> attains a global maximum at* (*t*, *<sup>x</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n, then*

$$-\psi\_l(t, \mathbf{x}) + \sup\_{\mathbf{u} \in \mathbf{U}} G(t, \mathbf{x}, \boldsymbol{\mu}, -\psi(t, \mathbf{x}), -\psi\_\mathbf{x}(t, \mathbf{x}), -\psi\_{\mathbf{x}\mathbf{x}}(t, \mathbf{x})) \le 0. \tag{24}$$

*(ii) A function v* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *is called a viscosity supersolution of (15) if v*(*T*, *<sup>x</sup>*) <sup>≥</sup> *<sup>h</sup>*(*x*), <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***n, and for any test function <sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*, whenever v* <sup>−</sup> *<sup>ψ</sup> attains a global minimum at* (*t*, *<sup>x</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n, then*

$$-\psi\_l(t, \mathbf{x}) + \sup\_{\mathbf{u} \in \mathbf{U}} G(t, \mathbf{x}, \boldsymbol{\mu}, -\psi(t, \mathbf{x}), -\psi\_\mathbf{x}(t, \mathbf{x}), -\psi\_{\mathbf{x}\mathbf{x}}(t, \mathbf{x})) \ge 0. \tag{25}$$

*(iii) If v* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *is both a viscosity subsolution and viscosity supersolution of (15), then it is called a viscosity solution of (15).*

In order to give the existence and uniqueness result for viscosity solution of the generalized HJB equation (15), it is convenient to give an intrinsic characterization of viscosity solutions. Let us recall the right parabolic super-subjets of a continuous function on [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* (see [18] or [3]). For *<sup>v</sup>* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) and (ˆ*t*, *<sup>x</sup>*ˆ) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n*, the *right parabolic superjet* of *<sup>v</sup>* at (ˆ*t*, *<sup>x</sup>*ˆ) is the set triple

$$\begin{split} \mathcal{P}\_{t+,\mathbf{x}}^{1,2,+} v(\mathbf{f}, \mathbf{\hat{x}}) &:= \left\{ (q, p, \mathbf{P}) \in \mathbf{R} \times \mathbf{R}^{\mathbf{n}} \times \mathcal{S}^{\mathbf{n}} \, \middle| \, v(t, \mathbf{x}) \le v(\mathbf{f}, \mathbf{\hat{x}}) + q(t - \mathbf{f}) + \langle p, \mathbf{x} - \mathbf{f} \rangle \\ &+ \frac{1}{2} (\mathbf{x} - \mathbf{\hat{x}})^{\top} P(\mathbf{x} - \mathbf{\hat{x}}) + o(|t - \mathbf{\hat{f}}| + |\mathbf{x} - \mathbf{\hat{x}}|^{2}), \text{ as } t \downarrow \mathbf{f}, \mathbf{x} \to \mathbf{\hat{x}} \right\}, \end{split} \tag{26}$$

and the *right parabolic subjet* of *v* at (ˆ*t*, *x*ˆ) is the set

$$\begin{split} \mathcal{P}\_{l+,\mathbf{x}}^{1,2-} v(\hat{t}, \mathbf{x}) &:= \left\{ (q, p, P) \in \mathbf{R} \times \mathbf{R}^{\mathrm{I}} \times \mathcal{S}^{\mathrm{II}} | v(t, \mathbf{x}) \ge v(\hat{t}, \mathbf{\hat{x}}) + q(t - \hat{t}) + \langle p, \mathbf{x} - \mathbf{\hat{x}} \rangle \\ &+ \frac{1}{2} (\mathbf{x} - \mathbf{\hat{x}})^{\top} P (\mathbf{x} - \mathbf{\hat{x}}) + o(|t - \hat{t}| + |\mathbf{x} - \mathbf{\hat{x}}|^{2}), \text{ as } t \downarrow \hat{t}, \mathbf{x} \to \mathbf{\hat{x}} \right\}. \end{split} \tag{27}$$

From the above definitions, we see immediately that

$$\begin{cases} \mathcal{P}^{1,2,+}\_{t+\mathcal{X}}v(\mathbf{f},\mathfrak{X}) + [0,\infty) \times \{0\} \times \mathcal{S}^{\eta}\_{+} = \mathcal{P}^{1,2,+}\_{t+\mathcal{X}}v(\mathbf{f},\mathfrak{X})\_{+},\\ \mathcal{P}^{1,2,-}\_{t+\mathcal{X}}v(\mathbf{f},\mathfrak{X}) - [0,\infty) \times \{0\} \times \mathcal{S}^{\eta}\_{+} = \mathcal{P}^{1,2,-}\_{t+\mathcal{X}}v(\mathbf{f},\mathfrak{X})\_{+} \end{cases}$$

Using the above definitions, we can give the following intrinsic formulation of viscosity

**Definition 3.2** *(i) A function v* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *satisfying v*(*T*, *<sup>x</sup>*) <sup>≤</sup> *<sup>h</sup>*(*x*), <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup> is a viscosity subsolution of (15) if, for any test function <sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*, if v* <sup>−</sup> *<sup>ψ</sup> attains a global maximum at*

*(ii) A function v* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *satisfying v*(*T*, *<sup>x</sup>*) <sup>≥</sup> *<sup>h</sup>*(*x*), <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup> is a viscosity supersolution of (15) if, for any test function <sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*, if v* <sup>−</sup> *<sup>ψ</sup> attains a global minimum at* (*t*, *<sup>x</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> and if* (*q*, *<sup>p</sup>*, *<sup>P</sup>*) ∈ P1,2,<sup>−</sup> *<sup>t</sup>*+,*<sup>x</sup> <sup>v</sup>*(*t*, *<sup>x</sup>*) *with q* <sup>=</sup> *<sup>ψ</sup>t*(*t*, *<sup>x</sup>*), *<sup>p</sup>* <sup>=</sup> *<sup>ψ</sup>x*(*t*, *<sup>x</sup>*), *<sup>P</sup>* <sup>≥</sup> *<sup>ψ</sup>xx*(*t*, *<sup>x</sup>*)*, then*

*(iii) If v* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *is both a viscosity subsolution and viscosity supersolution of (15), then it is*

*Proof* The result is immediate in view of Proposition 3.1. In fact it is a special case of

The following result is the existence and uniqueness of viscosity solution of the generalized

*(i) The value function V* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *defined by (6) is a unique viscosity solution of (15) in the*

*(ii) The value function V* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *is the only function that satisfies (12), (13) and the*

� , *x*�

� , *x*�

virtue of Propositions 3.1 and 3.2. The equivalence between (*i*) and (*ii*) is obvious.

*Proof* Result (*i*) is a special case of Theorems 3.1 and 4.1 of [14]. Result (*ii*) is obvious by

) > *v*(*t* � , *x*� ), ∀(*t* � , *x*�

) < *v*(*t* � , *x*� ), ∀(*t* � , *x*�

� <sup>≤</sup> 0, <sup>∀</sup>(*q*, *<sup>p</sup>*, *<sup>P</sup>*) ∈ P1,2,<sup>+</sup>

� <sup>≥</sup> 0, <sup>∀</sup>(*q*, *<sup>p</sup>*, *<sup>P</sup>*) ∈ P1,2,<sup>−</sup> *<sup>t</sup>*+,*<sup>x</sup> <sup>V</sup>*(*t*, *<sup>x</sup>*),

*<sup>t</sup>*+,*<sup>x</sup> V*(*t*, *x*),

) �= (*t*, *<sup>x</sup>*) <sup>∈</sup> [*t*, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*,

) �= (*t*, *<sup>x</sup>*) <sup>∈</sup> [*t*, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*,

**Theorem 3.1** *Suppose* **(H1)**∼**(H2)** *hold. Then we have the following equivalent results.*

*G*(*t*, *x*, *u*, −*ψ*(*t*, *x*), −*p*, −*P*)

*G*(*t*, *x*, *u*, −*ψ*(*t*, *x*), −*p*, −*P*)

*<sup>t</sup>*+,*<sup>x</sup> v*(*t*, *x*) *with q* = *ψt*(*t*, *x*), *p* = *ψx*(*t*, *x*), *P* ≤ *ψxx*(*t*, *x*)*,*

� <sup>≤</sup> 0. (32)

Stochastic Control for Jump Diff usions 129

� <sup>≥</sup> 0. (33)

solution of the generalized HJB equation (15).

(*t*, *<sup>x</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> and if* (*q*, *<sup>p</sup>*, *<sup>P</sup>*) ∈ P1,2,<sup>+</sup>

*called a viscosity solution of (15).*

*class of functions satisfying (12), (13).*

*following: For all* (*t*, *<sup>x</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n,*

Proposition 1 of [3].

HJB equation (15).

− *q* + sup *u*∈**U** �

− *q* + sup *u*∈**U** �

*V*(*T*, *x*) = *h*(*x*).

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

− *q* + sup *u*∈**U** �

− *q* + sup *u*∈**U** �

**Proposition 3.2** *Definitions 3.1 and 3.2 are equivalent.*

*G*(*t*, *x*, *u*, −*ψ*1(*t*, *x*), −*p*, −*P*)

*G*(*t*, *x*, *u*, −*ψ*2(*t*, *x*), −*p*, −*P*)

*<sup>ψ</sup>*<sup>1</sup> <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *such that <sup>ψ</sup>*1(*<sup>t</sup>*

*<sup>ψ</sup>*<sup>2</sup> <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *such that <sup>ψ</sup>*2(*<sup>t</sup>*

*then*

where <sup>S</sup>*<sup>n</sup>* <sup>+</sup> :<sup>=</sup> {*<sup>S</sup>* ∈ S*n*|*<sup>S</sup>* <sup>≥</sup> <sup>0</sup>}, and *<sup>A</sup>* <sup>±</sup> *<sup>B</sup>* :<sup>=</sup> {*<sup>a</sup>* <sup>±</sup> *<sup>b</sup>*|*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup>*, *<sup>b</sup>* <sup>∈</sup> *<sup>B</sup>*} for any subsets *<sup>A</sup>* and *<sup>B</sup>* in a same Euclidean space.

**Remark 3.1** Suppose that *<sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*). If *<sup>v</sup>* <sup>−</sup> *<sup>ψ</sup>* attains a global maximum at (ˆ*t*, *<sup>x</sup>*ˆ) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n*, then

$$\left(\psi\_t(\hat{t}, \hat{\mathfrak{x}}), \psi\_{\mathfrak{x}}(\hat{t}, \hat{\mathfrak{x}}), \psi\_{\mathfrak{x}\mathfrak{x}}(\hat{t}, \hat{\mathfrak{x}})\right) \in \mathcal{P}^{1,2,+}\_{t+\mathsf{x}} v(\hat{t}, \hat{\mathfrak{x}}).$$

If *<sup>v</sup>* <sup>−</sup> *<sup>ψ</sup>* attains a global minimum at (ˆ*t*, *<sup>x</sup>*ˆ) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n*, then

$$\left(\psi\_t(\mathfrak{f}, \mathfrak{k}), \psi\_{\mathfrak{x}}(\mathfrak{f}, \mathfrak{k}), \psi\_{\mathfrak{x}\mathfrak{x}}(\mathfrak{f}, \mathfrak{k})\right) \in \mathcal{P}^{1,2,-}\_{t+\mathsf{x}}\upsilon(\mathfrak{f}, \mathfrak{k}).$$

The following result is useful and whose proof for diffusion case can be found, for instance, in Lemma 5.4, Chapter 4 of [18].

**Proposition 3.1** *Let v* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *and* (*t*0, *<sup>x</sup>*0) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> be given. Then (i)* (*q*, *<sup>p</sup>*, *<sup>P</sup>*) ∈ P1,2,<sup>+</sup> *<sup>t</sup>*+,*<sup>x</sup> <sup>v</sup>*(*t*0, *<sup>x</sup>*0) *if and only if there exist a function <sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*, such that* � (*ψ*(*t*0, *<sup>x</sup>*0), *<sup>ψ</sup>t*(*t*0, *<sup>x</sup>*0), *<sup>ψ</sup>x*(*t*0, *<sup>x</sup>*0), *<sup>ψ</sup>xx*(*t*0, *<sup>x</sup>*0)) = (*v*(*t*0, *<sup>x</sup>*0), *<sup>q</sup>*, *<sup>p</sup>*, *<sup>P</sup>*), *<sup>ψ</sup>*(*t*, *<sup>x</sup>*) <sup>&</sup>gt; *<sup>v</sup>*(*t*, *<sup>x</sup>*), <sup>∀</sup>(*t*0, *<sup>x</sup>*0) �= (*t*, *<sup>x</sup>*) <sup>∈</sup> [*t*0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*. (28)

$$\begin{aligned} \text{(ii) } (q, p, P) \in \mathcal{P}\_{t+, \mathbf{x}}^{1, 2, -} v(t\_0, \mathbf{x}\_0) \text{ if and only if there exist a function } \boldsymbol{\psi} \in \mathbb{C}^{1,2}([0, T] \times \mathbb{R}^n), \text{ such that} \\\\ \begin{cases} (\boldsymbol{\psi}(t\_0, \mathbf{x}\_0), \boldsymbol{\psi}\_t(t\_0, \mathbf{x}\_0), \boldsymbol{\psi}\_x(t\_0, \mathbf{x}\_0), \boldsymbol{\psi}\_{\mathbf{x}}(t\_0, \mathbf{x}\_0)) = (v(t\_0, \mathbf{x}\_0), q, p, P), \\ \boldsymbol{\psi}(t, \mathbf{x}) < v(t, \mathbf{x}), \quad \boldsymbol{\forall}(t\_0, \mathbf{x}\_0) \neq (t, \mathbf{x}) \in [t\_0, T] \times \mathbb{R}^n. \end{aligned} \end{aligned} \tag{29}$$

We will also make use of the *partial* super-subjets with respect to one of the variables *t* and *x*. Therefore, we need the following definitions.

$$\begin{cases} \mathcal{P}^{2^{+}\_{\mathcal{X}}}v(\boldsymbol{\hat{t}},\boldsymbol{\hat{x}}) := \left\{ (p,P) \in \times \mathbb{R}^{n} \times \mathcal{S}^{n} \, | \, v(\boldsymbol{\hat{t}},\boldsymbol{x}) \le v(\boldsymbol{\hat{t}},\boldsymbol{\hat{x}}) + \langle p,\boldsymbol{x}-\boldsymbol{\hat{x}} \rangle \right. \\ \qquad \quad + \frac{1}{2}(\mathbf{x}-\boldsymbol{\hat{x}})^{\top}P(\mathbf{x}-\boldsymbol{\hat{x}}) + o(|\mathbf{x}-\boldsymbol{\hat{x}}|^{2}), \text{ as } \mathbf{x} \to \mathbf{\hat{x}} \right\}, \\ \mathcal{P}^{2^{-}\_{\mathcal{X}}}v(\boldsymbol{\hat{t}},\boldsymbol{\hat{x}}) := \left\{ (p,P) \in \times \mathbb{R}^{n} \times \mathcal{S}^{n} \, | \, v(\boldsymbol{\hat{t}},\boldsymbol{x}) \ge v(\boldsymbol{\hat{t}},\boldsymbol{\hat{x}}) + \langle p,\boldsymbol{x}-\boldsymbol{\hat{x}} \rangle \right. \\ \qquad \quad + \frac{1}{2}(\mathbf{x}-\boldsymbol{\hat{x}})^{\top}P(\mathbf{x}-\boldsymbol{\hat{x}}) + o(|\mathbf{x}-\boldsymbol{\hat{x}}|^{2}), \text{ as } \mathbf{x} \to \mathbf{\hat{x}} \right\}, \end{cases} \tag{30}$$

and

$$\begin{cases} \mathcal{P}\_{t+}^{1,+} v(\mathbf{f}, \boldsymbol{\mathfrak{t}}) := \left\{ q \in \mathbb{R} \, \middle| \, v(t, \boldsymbol{\mathfrak{t}}) \le v(\boldsymbol{\mathfrak{f}}, \boldsymbol{\mathfrak{t}}) + q(t - \boldsymbol{\mathfrak{f}}) + o(|t - \boldsymbol{\mathfrak{f}}|), \text{ as } t \downarrow \boldsymbol{\mathfrak{t}} \right\}, \\\mathcal{P}\_{t+}^{1,-} v(\boldsymbol{\mathfrak{t}}, \boldsymbol{\mathfrak{t}}) := \left\{ q \in \mathbb{R} \, \middle| \, v(t, \boldsymbol{\mathfrak{t}}) \ge v(\boldsymbol{\mathfrak{f}}, \boldsymbol{\mathfrak{t}}) + q(t - \boldsymbol{\mathfrak{f}}) + o(|t - \boldsymbol{\mathfrak{f}}|), \text{ as } t \downarrow \boldsymbol{\mathfrak{t}} \right\}. \end{cases} \tag{31}$$

Using the above definitions, we can give the following intrinsic formulation of viscosity solution of the generalized HJB equation (15).

**Definition 3.2** *(i) A function v* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *satisfying v*(*T*, *<sup>x</sup>*) <sup>≤</sup> *<sup>h</sup>*(*x*), <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup> is a viscosity subsolution of (15) if, for any test function <sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*, if v* <sup>−</sup> *<sup>ψ</sup> attains a global maximum at* (*t*, *<sup>x</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> and if* (*q*, *<sup>p</sup>*, *<sup>P</sup>*) ∈ P1,2,<sup>+</sup> *<sup>t</sup>*+,*<sup>x</sup> v*(*t*, *x*) *with q* = *ψt*(*t*, *x*), *p* = *ψx*(*t*, *x*), *P* ≤ *ψxx*(*t*, *x*)*, then*

$$-q + \sup\_{\mu \in \mathbf{U}} \left\{ G(t, \mathbf{x}, \mu\_\prime - \psi(t, \mathbf{x}), -p\_\prime - P) \right\} \le 0. \tag{32}$$

*(ii) A function v* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *satisfying v*(*T*, *<sup>x</sup>*) <sup>≥</sup> *<sup>h</sup>*(*x*), <sup>∀</sup>*<sup>x</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup> is a viscosity supersolution of (15) if, for any test function <sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*, if v* <sup>−</sup> *<sup>ψ</sup> attains a global minimum at* (*t*, *<sup>x</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> and if* (*q*, *<sup>p</sup>*, *<sup>P</sup>*) ∈ P1,2,<sup>−</sup> *<sup>t</sup>*+,*<sup>x</sup> <sup>v</sup>*(*t*, *<sup>x</sup>*) *with q* <sup>=</sup> *<sup>ψ</sup>t*(*t*, *<sup>x</sup>*), *<sup>p</sup>* <sup>=</sup> *<sup>ψ</sup>x*(*t*, *<sup>x</sup>*), *<sup>P</sup>* <sup>≥</sup> *<sup>ψ</sup>xx*(*t*, *<sup>x</sup>*)*, then*

$$-q + \sup\_{\mu \in \mathbf{U}} \left\{ G(t, \mathbf{x}, \mu, -\psi(t, \mathbf{x}), -p\_\prime - P) \right\} \ge 0. \tag{33}$$

*(iii) If v* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *is both a viscosity subsolution and viscosity supersolution of (15), then it is called a viscosity solution of (15).*

**Proposition 3.2** *Definitions 3.1 and 3.2 are equivalent.*

10 Will-be-set-by-IN-TECH

<sup>+</sup> :<sup>=</sup> {*<sup>S</sup>* ∈ S*n*|*<sup>S</sup>* <sup>≥</sup> <sup>0</sup>}, and *<sup>A</sup>* <sup>±</sup> *<sup>B</sup>* :<sup>=</sup> {*<sup>a</sup>* <sup>±</sup> *<sup>b</sup>*|*<sup>a</sup>* <sup>∈</sup> *<sup>A</sup>*, *<sup>b</sup>* <sup>∈</sup> *<sup>B</sup>*} for any subsets *<sup>A</sup>* and *<sup>B</sup>* in

�

�

*<sup>t</sup>*+,*<sup>x</sup> <sup>v</sup>*(*t*0, *<sup>x</sup>*0) *if and only if there exist a function <sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*, such that*

*<sup>ψ</sup>*(*t*, *<sup>x</sup>*) <sup>&</sup>gt; *<sup>v</sup>*(*t*, *<sup>x</sup>*), <sup>∀</sup>(*t*0, *<sup>x</sup>*0) �= (*t*, *<sup>x</sup>*) <sup>∈</sup> [*t*0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*. (28)

*<sup>ψ</sup>*(*t*, *<sup>x</sup>*) <sup>&</sup>lt; *<sup>v</sup>*(*t*, *<sup>x</sup>*), <sup>∀</sup>(*t*0, *<sup>x</sup>*0) �= (*t*, *<sup>x</sup>*) <sup>∈</sup> [*t*0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*. (29)

�*v*(ˆ*t*, *<sup>x</sup>*) <sup>≤</sup> *<sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ) + �*p*, *<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*ˆ�

�*v*(ˆ*t*, *<sup>x</sup>*) <sup>≥</sup> *<sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ) + �*p*, *<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*ˆ�

<sup>2</sup>), as *<sup>x</sup>* <sup>→</sup> *<sup>x</sup>*<sup>ˆ</sup>

<sup>2</sup>), as *<sup>x</sup>* <sup>→</sup> *<sup>x</sup>*<sup>ˆ</sup>

� ,

(30)

(31)

� ,

> � ,

> � .

∈ P1,2,<sup>+</sup>

**Remark 3.1** Suppose that *<sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*). If *<sup>v</sup>* <sup>−</sup> *<sup>ψ</sup>* attains a global maximum at (ˆ*t*, *<sup>x</sup>*ˆ) <sup>∈</sup>

The following result is useful and whose proof for diffusion case can be found, for instance,

� (*ψ*(*t*0, *<sup>x</sup>*0), *<sup>ψ</sup>t*(*t*0, *<sup>x</sup>*0), *<sup>ψ</sup>x*(*t*0, *<sup>x</sup>*0), *<sup>ψ</sup>xx*(*t*0, *<sup>x</sup>*0)) = (*v*(*t*0, *<sup>x</sup>*0), *<sup>q</sup>*, *<sup>p</sup>*, *<sup>P</sup>*),

*(ii)* (*q*, *<sup>p</sup>*, *<sup>P</sup>*) ∈ P1,2,<sup>−</sup> *<sup>t</sup>*+,*<sup>x</sup> <sup>v</sup>*(*t*0, *<sup>x</sup>*0) *if and only if there exist a function <sup>ψ</sup>* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*, such that*

� (*ψ*(*t*0, *<sup>x</sup>*0), *<sup>ψ</sup>t*(*t*0, *<sup>x</sup>*0), *<sup>ψ</sup>x*(*t*0, *<sup>x</sup>*0), *<sup>ψ</sup>xx*(*t*0, *<sup>x</sup>*0)) = (*v*(*t*0, *<sup>x</sup>*0), *<sup>q</sup>*, *<sup>p</sup>*, *<sup>P</sup>*),

We will also make use of the *partial* super-subjets with respect to one of the variables *t* and *x*.

(*x* − *x*ˆ)�*P*(*x* − *x*ˆ) + *o*(|*x* − *x*ˆ|

(*x* − *x*ˆ)�*P*(*x* − *x*ˆ) + *o*(|*x* − *x*ˆ|

�*v*(*t*, *<sup>x</sup>*ˆ) <sup>≤</sup> *<sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ) + *<sup>q</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>ˆ</sup>*t*) + *<sup>o</sup>*(|*<sup>t</sup>* <sup>−</sup> <sup>ˆ</sup>*t*|), as *<sup>t</sup>* <sup>↓</sup> <sup>ˆ</sup>*<sup>t</sup>*

�*v*(*t*, *<sup>x</sup>*ˆ) <sup>≥</sup> *<sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ) + *<sup>q</sup>*(*<sup>t</sup>* <sup>−</sup> <sup>ˆ</sup>*t*) + *<sup>o</sup>*(|*<sup>t</sup>* <sup>−</sup> <sup>ˆ</sup>*t*|), as *<sup>t</sup>* <sup>↓</sup> <sup>ˆ</sup>*<sup>t</sup>*

(*p*, *<sup>P</sup>*) ∈ ×**R***<sup>n</sup>* × S*n*�

(*p*, *<sup>P</sup>*) ∈ ×**R***<sup>n</sup>* × S*n*�

<sup>+</sup> <sup>=</sup> <sup>P</sup>1,2,<sup>+</sup>

*<sup>t</sup>*+,*<sup>x</sup> <sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ),

<sup>+</sup> <sup>=</sup> <sup>P</sup>1,2,<sup>−</sup> *<sup>t</sup>*+,*<sup>x</sup> <sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ),

*<sup>t</sup>*+,*<sup>x</sup> <sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ).

∈ P1,2,<sup>−</sup> *<sup>t</sup>*+,*<sup>x</sup> <sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ).

*<sup>t</sup>*+,*<sup>x</sup> <sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ)+[0, <sup>∞</sup>) × {0}×S*<sup>n</sup>*

<sup>P</sup>1,2,<sup>−</sup> *<sup>t</sup>*+,*<sup>x</sup> <sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ) <sup>−</sup> [0, <sup>∞</sup>) × {0}×S*<sup>n</sup>*

*ψt*(ˆ*t*, *x*ˆ), *ψx*(ˆ*t*, *x*ˆ), *ψxx*(ˆ*t*, *x*ˆ)

*ψt*(ˆ*t*, *x*ˆ), *ψx*(ˆ*t*, *x*ˆ), *ψxx*(ˆ*t*, *x*ˆ)

**Proposition 3.1** *Let v* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *and* (*t*0, *<sup>x</sup>*0) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> be given. Then*

From the above definitions, we see immediately that

P1,2,<sup>+</sup>

�

�

Therefore, we need the following definitions.

*<sup>x</sup> v*(ˆ*t*, *x*ˆ) :=

*<sup>x</sup> v*(ˆ*t*, *x*ˆ) :=

�

+ 1 2

+ 1 2

� *q* ∈ **R** �

� *q* ∈ **R** �

�

⎧

<sup>P</sup>2,<sup>+</sup>

<sup>P</sup>2,<sup>−</sup>

*<sup>t</sup>*<sup>+</sup> *<sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ) :=

*<sup>t</sup>*<sup>+</sup> *<sup>v</sup>*(ˆ*t*, *<sup>x</sup>*ˆ) :=

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

P1,<sup>+</sup>

P1,<sup>−</sup>

⎧ ⎪⎨

⎪⎩

and

If *<sup>v</sup>* <sup>−</sup> *<sup>ψ</sup>* attains a global minimum at (ˆ*t*, *<sup>x</sup>*ˆ) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n*, then

⎧ ⎨ ⎩

where <sup>S</sup>*<sup>n</sup>*

a same Euclidean space.

in Lemma 5.4, Chapter 4 of [18].

[0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n*, then

*(i)* (*q*, *<sup>p</sup>*, *<sup>P</sup>*) ∈ P1,2,<sup>+</sup>

*Proof* The result is immediate in view of Proposition 3.1. In fact it is a special case of Proposition 1 of [3].

The following result is the existence and uniqueness of viscosity solution of the generalized HJB equation (15).

**Theorem 3.1** *Suppose* **(H1)**∼**(H2)** *hold. Then we have the following equivalent results.*

*(i) The value function V* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *defined by (6) is a unique viscosity solution of (15) in the class of functions satisfying (12), (13).*

*(ii) The value function V* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *is the only function that satisfies (12), (13) and the following: For all* (*t*, *<sup>x</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***n,*

$$\begin{cases} \begin{aligned} & -q + \sup\_{u \in \mathcal{U}} \left\{ G(t, x, \mu, -\psi\_1(t, x), -p, -P) \right\} \le 0, \quad \forall (q, p, P) \in \mathcal{P}^{1, 2, +}\_{t + \mathcal{X}} V(t, x), \\ & \quad \forall p \in \mathcal{C}^{1, 2}([0, T] \times \mathbb{R}^n) \text{ such that } \psi\_1(t', x') > v(t', x'), \; \forall (t', x') \ne (t, x) \in [t, T] \times \mathbb{R}^n, \\ & -q + \sup\_{u \in \mathcal{U}} \left\{ G(t, x, u, -\psi\_2(t, x), -p, -P) \right\} \ge 0, \quad \forall (q, p, P) \in \mathcal{P}^{1, 2, -}\_{t + \mathcal{X}} V(t, x), \\ & \quad \forall p \in \mathcal{C}^{1, 2}([0, T] \times \mathbb{R}^n) \text{ such that } \psi\_2(t', x') < v(t', x'), \; \forall (t', x') \ne (t, x) \in [t, T] \times \mathbb{R}^n, \\ & V(T, x) = h(\mathbf{x}). \end{aligned} \right\}$$

*Proof* Result (*i*) is a special case of Theorems 3.1 and 4.1 of [14]. Result (*ii*) is obvious by virtue of Propositions 3.1 and 3.2. The equivalence between (*i*) and (*ii*) is obvious.

#### **3.2. Main results: Relationship between Stochastic MP and DPP**

Proposition 2.1 tell us that the value function has nice continuity properties. But in general we cannot obtain the differentiablity of it. Therefore we should not suppose the generalized HJB equation (15) always admits an enough smooth (classic) solution. In fact it is not true even in the simplest case; see Example 3.2 in this subsection. This is an important *gap* in the literature (see Section 2, [7], for example). Fortunately, this gap can be bridged by the theory of viscosity solutions. This is one of the main contributions of this paper.

The following result shows that the adjoint process *p*, *P* and the value function *V* relate to each other within the framework of the superjet and the subjet in the state variable *x* along an optimal trajectory.

**Theorem 3.2** *Suppose* **(H1)**∼**(H3)** *hold and let* (*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> be fixed. Let* (*x*¯*s*,*y*;*u*¯ (·), *u*¯(·)) *be an optimal pair of our stochastic optimal control problem. Let* (*p*(·), *q*(·), *γ*(·, ·)) *and* (*P*(·), *Q*(·), *R*(·, ·)) *are first-order and second-order adjoint processes, respectively. Then*

$$\{-p(t)\} \times [-P(t), \infty) \subseteq \mathcal{P}\_{\text{x}}^{\mathcal{Q}, +} V(t, \check{\mathbf{x}}^{\mathcal{A}, \mathcal{Y}^{\mathcal{A}}}(t)), \quad \forall t \in [\mathbf{s}, T], \quad \mathbf{P}\text{-a.s.},\tag{34}$$

where

and

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

where ⎧ ⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

and

⎧ ⎪⎪⎪⎪⎪⎪⎪⎨

*εz*1(*r*) :=

*εz*2(*r*) :=

*εz*3(*r*, ·) :=

(*r*) = � ¯ *bx*(*r*)*ξt*,*z*;*u*¯

> + �

+ � **E** �

(*t*) = *<sup>z</sup>* <sup>−</sup> *<sup>x</sup>s*,*y*;*u*¯

(<sup>1</sup> <sup>−</sup> *<sup>θ</sup>*)*ξt*,*z*;*u*¯

(<sup>1</sup> <sup>−</sup> *<sup>θ</sup>*)*ξt*,*z*;*u*¯

(<sup>1</sup> <sup>−</sup> *<sup>θ</sup>*)*ξt*,*z*;*u*¯

*<sup>δ</sup>* : [0, <sup>∞</sup>) <sup>→</sup> [0, <sup>∞</sup>), independent of *<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***n*, with *<sup>δ</sup>*(*r*)







� 1 0 � *bx*(*r*, *x*¯

� 1 0 � *σx*(*r*, *x*¯

� 1 0 � *cx*(*r*, *x*¯

*s*,*y*;*u*¯

*s*,*y*;*u*¯

*s*,*y*;*u*¯

(*r*) + <sup>1</sup> 2 *ξt*,*z*;*u*¯

> (*r*) + <sup>1</sup> 2 *ξt*,*z*;*u*¯

<sup>+</sup> *<sup>ε</sup>z*4(*r*)*dr* <sup>+</sup> *<sup>ε</sup>z*5(*r*)*dW*(*r*) + �

*bxx*(*r*, *x*¯

*σxx*(*r*, *x*¯

*cxx*(*r*, *x*¯

*σ*¯*x*(*r*)*ξt*,*z*;*u*¯

*c*¯*x*(*r*,*e*)*ξt*,*z*;*u*¯

(*t*),

(*r*)��

(*r*)��

(*r*−)��

<sup>2</sup>*dr*� �F*s t* �

<sup>2</sup>*dr*� �F*s t* �

<sup>2</sup>*dr*� �F*s t* �

<sup>2</sup>*dr*� �F*s t* �

<sup>L</sup><sup>2</sup> *dr*� �F*s t* �

<sup>L</sup><sup>2</sup> *dr*� �F*s t* �

(*r*) + *θξt*,*z*;*u*¯

(*r*) + *θξt*,*z*;*u*¯

(*r*−) + *θξt*,*z*;*u*¯

(*r*−) + <sup>1</sup> 2 *ξt*,*z*;*u*¯

*s*,*y*;*u*¯

*s*,*y*;*u*¯

*s*,*y*;*u*¯

We are going to show that, there exists a deterministic continuous and increasing function

(*ω*) ≤ *δ*(|*z* − *x*¯

(*ω*) ≤ *δ*(|*z* − *x*¯

(*ω*) ≤ *δ*(|*z* − *x*¯

(*ω*) ≤ *δ*(|*z* − *x*¯

(*ω*) ≤ *δ*(|*z* − *x*¯

(*ω*) ≤ *δ*(|*z* − *x*¯

(*r*)� ¯

(*r*), *<sup>u</sup>*¯(*r*)) <sup>−</sup> ¯

*bxx*(*r*)*ξt*,*z*;*u*¯

(*r*)�*σ*¯*xx*(*r*)*ξt*,*z*;*u*¯

**E**

(*r*) + *θξt*,*z*;*u*¯

(*r*) + *θξt*,*z*;*u*¯

(*r*−) + *θξt*,*z*;*u*¯

*s*,*y*;*u*¯

*s*,*y*;*u*¯

*s*,*y*;*u*¯

*s*,*y*;*u*¯

(*r*), *u*¯(*r*)) − *σ*¯*x*(*r*)

*bx*(*r*) � *ξt*,*z*;*u*¯

(*r*−), *u*¯(*r*), ·) − *c*¯*x*(*r*, ·)

(*r*) � *dr*

> (*r*) � *dW*(*r*)

(*r*−)�*c*¯*xx*(*r*,*e*)*ξt*,*z*;*u*¯

*<sup>ε</sup>z*6(*r*,*e*)*N*˜ (*dedr*), *<sup>r</sup>* <sup>∈</sup> [*t*, *<sup>T</sup>*],

(*r*), *<sup>u</sup>*¯(*r*)) <sup>−</sup> ¯

*<sup>r</sup>* → 0 as *r* → 0, such that

(*t*, *ω*)|

(*t*, *ω*)|

(*t*, *ω*)|

(*t*, *ω*)|

*s*,*y*;*u*¯

(*t*, *ω*)|

(*t*, *ω*)|

*s*,*y*;*u*¯

(*r*), *u*¯(*r*)) − *σ*¯*xx*(*r*)

� *ξt*,*z*;*u*¯

(*r*)*dθ*,

(*r*)*dθ*,

(*r*−)*dθ*,

Stochastic Control for Jump Diff usions 131

*N*˜ (*dedr*)

(*r*)*dθ*,

(*r*)*dθ*,

� *ξt*,*z*;*u*¯ (40)

(*r*−)*dθ*.

(41)

(42)

� *ξt*,*z*;*u*¯

(*r*−) �

*bxx*(*r*) � *ξt*,*z*;*u*¯

(*r*−), *u*¯(*r*), ·) − *c*¯*xx*(*r*, ·)

<sup>2</sup>), **P**-*a*.*s*.*ω*,

<sup>2</sup>), **P**-*a*.*s*.*ω*,

<sup>4</sup>), **P**-*a*.*s*.*ω*,

<sup>4</sup>), **P**-*a*.*s*.*ω*,

<sup>2</sup>), **P**-*a*.*s*.*ω*,

<sup>4</sup>), **P**-*a*.*s*.*ω*.

� *ξt*,*z*;*u*¯

⎪⎪⎪⎪⎪⎪⎪⎩

*dξt*,*z*;*u*¯

*ξt*,*z*;*u*¯

� 1 0

� 1 0

� 1 0

⎧ ⎪⎪⎪⎪⎪⎪⎨ **E** � � *<sup>T</sup> t*

**E** � � *<sup>T</sup> t*

**E** � � *<sup>T</sup> t*

**E** � � *<sup>T</sup> t*

**E** � � *<sup>T</sup> t*

**E** � � *<sup>T</sup> t*

⎪⎪⎪⎪⎪⎪⎩

⎧ ⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

*εz*4(*r*) :=

*εz*5(*r*) :=

*εz*6(*r*, ·) :=

$$\mathcal{P}\_{\boldsymbol{x}}^{2-} \boldsymbol{V}(t, \boldsymbol{\tilde{x}}^{\boldsymbol{y}, \boldsymbol{y}; \boldsymbol{\tilde{a}}}(t)) \subseteq \{-\boldsymbol{p}(t)\} \times (-\infty, \boldsymbol{P}(t)], \quad \forall t \in [\boldsymbol{s}, T], \quad \mathbf{P}\text{-}\boldsymbol{a}. \tag{35}$$

*We also have*

$$\mathcal{P}\_{\boldsymbol{x}}^{1,-}V(t,\boldsymbol{\tilde{x}}^{\boldsymbol{s},\boldsymbol{y};\boldsymbol{\mathcal{R}}}(t)) \subseteq \{-p(t)\} \subseteq \mathcal{P}\_{\boldsymbol{x}}^{1,+}V(t,\boldsymbol{\tilde{x}}^{\boldsymbol{s},\boldsymbol{y};\boldsymbol{\mathcal{R}}}(t)), \quad \forall t \in [\boldsymbol{s},T], \quad \mathbf{P}\text{-a.s.}\tag{36}$$

*Proof* Fix a *<sup>t</sup>* <sup>∈</sup> [*s*, *<sup>T</sup>*]. For any *<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***n*, denote by *<sup>x</sup>t*,*z*;*u*¯ (·) the solution of the following SDEP:

$$\begin{split} \mathbf{x}^{t; \overline{z}; \underline{\boldsymbol{\mu}}}(\boldsymbol{r}) &= \mathbf{z} + \int\_{t}^{\boldsymbol{r}} \boldsymbol{b}(\boldsymbol{r}, \mathbf{x}^{t; \overline{z}; \boldsymbol{\mu}}(\boldsymbol{r}), \overline{\boldsymbol{u}}(\boldsymbol{r})) d\boldsymbol{r} + \int\_{t}^{\boldsymbol{r}} \boldsymbol{\sigma}(\boldsymbol{r}, \mathbf{x}^{t; \overline{z}; \boldsymbol{\mu}}(\boldsymbol{r}), \overline{\boldsymbol{u}}(\boldsymbol{r})) d\boldsymbol{W}(\boldsymbol{r}) \\ &+ \int\_{\mathbf{E}} \int\_{t}^{\boldsymbol{r}} \boldsymbol{c}(\boldsymbol{r}, \mathbf{x}^{t; \overline{z}; \boldsymbol{\mu}}(\boldsymbol{r} - \boldsymbol{)}, \overline{\boldsymbol{u}}(\boldsymbol{r}), \boldsymbol{e}) \tilde{\boldsymbol{N}}(\boldsymbol{d} \boldsymbol{d} \boldsymbol{r}), \quad \boldsymbol{r} \in [t, T]. \end{split} \tag{37}$$

It is clear that (37) can be regarded as an SDEP on � <sup>Ω</sup>, <sup>F</sup>, {F*<sup>s</sup> <sup>r</sup>* }*r*≥*s*, **<sup>P</sup>**(·|F*<sup>s</sup> <sup>t</sup>* )(*ω*) � for **P**-*a*.*s*.*ω*, where **<sup>P</sup>**(·|F*<sup>s</sup> <sup>t</sup>* )(*ω*) is the regular conditional probability given <sup>F</sup>*<sup>s</sup> <sup>t</sup>* defined on (Ω, F) (see pp. 12-16 of [10]). In probability space (Ω, <sup>F</sup>, **<sup>P</sup>**(·|F*<sup>s</sup> <sup>t</sup>* )(*ω*)), random variable *<sup>x</sup>*¯*s*,*y*;*u*¯ (*t*, *ω*) is almost surely a constant vector in **R***<sup>n</sup>* (we still denote it by *x*¯*s*,*y*;*u*¯ (*t*, *ω*)).

$$\text{Set } \xi^{t, z; \bar{\mu}}(r) := x^{t, z; \bar{\mu}}(r) - \bar{x}^{t, \bar{\mu}^{s, \bar{\mu}}(t); \bar{\mu}}(r), t \le r \le T. \text{ Thus by Lemma 2.1 we have}$$

$$\mathbb{E}\left[\sup\_{t\le r\le T}|\xi^{t,\overline{z};\overline{\mu}}(r)|^{k}|\mathcal{F}\_{l}^{\overline{s}}\right](\omega) \le \mathbb{C}|z-\overline{x}^{\mathrm{s},y;\overline{\mu}}(t,\omega)|^{k}, \quad \mathbb{P}\text{-a.s.}\omega, \quad \text{for } k=2,4. \tag{38}$$

Now we rewrite the equation for *ξt*,*z*;*u*¯ (·) in two different ways based on different orders of expansion, which called the *first-order and second-order variational equations*, respectively:

$$\begin{cases} d\xi^{t,z;\overline{\mu}}(r) = \bar{b}\_{\text{X}}(r)\xi^{t,z;\overline{\mu}}(r)dr + \sigma\_{\text{X}}(r)\xi^{t,z;\overline{\mu}}(r)dW(r) + \int\_{\mathbb{E}} \varepsilon\_{\text{X}}(r,e)\xi^{t,z;\overline{\mu}}(r-)\hat{N}(dedr) \\\\ \quad + \varepsilon\_{\text{z1}}(r)dr + \varepsilon\_{\text{z2}}(r)dW(r) + \int\_{\mathbb{E}} \varepsilon\_{\text{z3}}(r,e)\hat{N}(dedr), \quad r \in [t,T], \end{cases} \tag{39}$$
  $\xi^{t,z;\overline{\mu}}(t) = z - \xi^{s,y;\overline{\mu}}(t),$ 

where

12 Will-be-set-by-IN-TECH

Proposition 2.1 tell us that the value function has nice continuity properties. But in general we cannot obtain the differentiablity of it. Therefore we should not suppose the generalized HJB equation (15) always admits an enough smooth (classic) solution. In fact it is not true even in the simplest case; see Example 3.2 in this subsection. This is an important *gap* in the literature (see Section 2, [7], for example). Fortunately, this gap can be bridged by the theory of viscosity

The following result shows that the adjoint process *p*, *P* and the value function *V* relate to each other within the framework of the superjet and the subjet in the state variable *x* along an

*be an optimal pair of our stochastic optimal control problem. Let* (*p*(·), *q*(·), *γ*(·, ·)) *and*

*s*,*y*;*u*¯

*<sup>x</sup> V*(*t*, *x*¯

(*r*), *u*¯(*r*))*dr* +

*s*,*y*;*u*¯

� *r t*

(*r*−), *<sup>u</sup>*¯(*r*),*e*)*N*˜ (*dedr*), *<sup>r</sup>* <sup>∈</sup> [*t*, *<sup>T</sup>*].

<sup>Ω</sup>, <sup>F</sup>, {F*<sup>s</sup>*

(*t*, *ω*)).

(*r*), *t* ≤ *r* ≤ *T*. Thus by Lemma 2.1 we have

� **E**

(*t*, *ω*)|

(*r*)*dW*(*r*) +

*s*,*y*;*u*¯

*<sup>t</sup>* )(*ω*)), random variable *<sup>x</sup>*¯*s*,*y*;*u*¯

(·) in two different ways based on different orders of

*c*¯*x*(*r*,*e*)*ξt*,*z*;*u*¯

*<sup>ε</sup>z*3(*r*,*e*)*N*˜ (*dedr*), *<sup>r</sup>* <sup>∈</sup> [*t*, *<sup>T</sup>*],

(*t*)) ⊆ {−*p*(*t*)} × (−∞, *P*(*t*)], ∀*t* ∈ [*s*, *T*], **P***-a*.*s*. (35)

*σ*(*r*, *xt*,*z*;*u*¯

*<sup>x</sup> V*(*t*, *x*¯

**Theorem 3.2** *Suppose* **(H1)**∼**(H3)** *hold and let* (*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> be fixed. Let* (*x*¯*s*,*y*;*u*¯

(*P*(·), *Q*(·), *R*(·, ·)) *are first-order and second-order adjoint processes, respectively. Then*

(*t*)) ⊆ {−*p*(*t*)}⊆P1,<sup>+</sup>

*b*(*r*, *xt*,*z*;*u*¯

*c*(*r*, *xt*,*z*;*u*¯

*<sup>t</sup>* )(*ω*) is the regular conditional probability given <sup>F</sup>*<sup>s</sup>*

(*ω*) ≤ *C*|*z* − *x*¯

(*r*)*dr* + *σ*¯*x*(*r*)*ξt*,*z*;*u*¯

+ *εz*1(*r*)*dr* + *εz*2(*r*)*dW*(*r*) +

expansion, which called the *first-order and second-order variational equations*, respectively:

� **E**

**3.2. Main results: Relationship between Stochastic MP and DPP**

solutions. This is one of the main contributions of this paper.

{−*p*(*t*)} × [−*P*(*t*), <sup>∞</sup>) ⊆ P2,<sup>+</sup>

*s*,*y*;*u*¯

*Proof* Fix a *<sup>t</sup>* <sup>∈</sup> [*s*, *<sup>T</sup>*]. For any *<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***n*, denote by *<sup>x</sup>t*,*z*;*u*¯

� *r t*

*s*,*y*;*u*¯

(*r*) = *z* +

+ � **E** � *r t*

12-16 of [10]). In probability space (Ω, <sup>F</sup>, **<sup>P</sup>**(·|F*<sup>s</sup>*

(*r*) <sup>−</sup> *<sup>x</sup>*¯*t*,*x*¯*s*,*y*;*u*¯

<sup>|</sup>*ξt*,*z*;*u*¯ (*r*)| *<sup>k</sup>*|F*<sup>s</sup> t* �

*bx*(*r*)*ξt*,*z*;*u*¯

*s*,*y*;*u*¯ (*t*),

Now we rewrite the equation for *ξt*,*z*;*u*¯

(*r*) = ¯

(*t*) = *z* − *x*¯

It is clear that (37) can be regarded as an SDEP on �

surely a constant vector in **R***<sup>n</sup>* (we still denote it by *x*¯*s*,*y*;*u*¯

(*t*);*u*¯

optimal trajectory.

*We also have*

where **<sup>P</sup>**(·|F*<sup>s</sup>*

Set *ξt*,*z*;*u*¯

⎧ ⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

<sup>P</sup>2,<sup>−</sup> *<sup>x</sup> V*(*t*, *x*¯

<sup>P</sup>1,<sup>−</sup> *<sup>x</sup> V*(*t*, *x*¯

*xt*,*z*;*u*¯

(*r*) := *xt*,*z*;*u*¯

**E** � sup *t*≤*r*≤*T*

*dξt*,*z*;*u*¯

*ξt*,*z*;*u*¯

$$\begin{cases} \begin{aligned} \varepsilon\_{z1}(r) &:= \int\_{0}^{1} \left[ b\_{\mathcal{X}}(r, \bar{\mathbf{x}}^{s,y;\mathcal{R}}(r) + \theta\_{\mathsf{S}}^{\mathsf{zt},z;\mathcal{R}}(r), \bar{u}(r)) - \bar{b}\_{\mathcal{X}}(r) \right] \xi^{\mathsf{tz},z;\mathcal{R}}(r) d\theta, \\\ \varepsilon\_{z2}(r) &:= \int\_{0}^{1} \left[ \sigma\_{\mathcal{X}}(r, \bar{\mathbf{x}}^{s,y;\mathcal{R}}(r) + \theta\_{\mathsf{S}}^{\mathsf{zt},z;\mathcal{R}}(r), \bar{u}(r)) - \bar{\sigma}\_{\mathcal{X}}(r) \right] \xi^{\mathsf{tz},z;\mathcal{R}}(r) d\theta, \\\ \varepsilon\_{z3}(r, \cdot) &:= \int\_{0}^{1} \left[ c\_{\mathcal{X}}(r, \bar{\mathbf{x}}^{s,y;\mathcal{R}}(r-) + \theta\_{\mathsf{S}}^{\mathsf{zt},z;\mathcal{R}}(r-), \bar{u}(r), \cdot) - \bar{\sigma}\_{\mathcal{X}}(r\_{\prime} \cdot) \right] \xi^{\mathsf{t},z;\mathcal{R}}(r-) d\theta, \end{aligned} \end{cases}$$

and

(·), *u*¯(·))

(37)

(39)

for **P**-*a*.*s*.*ω*,

(*t*, *ω*) is almost

(*t*)), ∀*t* ∈ [*s*, *T*], **P***-a*.*s*., (34)

(*t*)), ∀*t* ∈ [*s*, *T*], **P***-a*.*s*. (36)

(·) the solution of the following SDEP:

(*r*), *u*¯(*r*))*dW*(*r*)

*<sup>t</sup>* )(*ω*) �

*<sup>k</sup>*, **P**-*a*.*s*.*ω*., for *k* = 2, 4. (38)

*<sup>t</sup>* defined on (Ω, F) (see pp.

(*r*−)*N*˜ (*dedr*)

*<sup>r</sup>* }*r*≥*s*, **<sup>P</sup>**(·|F*<sup>s</sup>*

$$\begin{cases} d\xi^{t,z;\mathfrak{a}}(r) = \left\{\bar{b}\_{\mathbf{x}}(r)\xi^{t,z;\mathfrak{a}}(r) + \frac{1}{2}\xi^{t,z;\mathfrak{a}}(r)\,^\top\bar{b}\_{\mathbf{x}\mathbf{x}}(r)\xi^{t,z;\mathfrak{a}}(r)\right\} dr \\ \qquad + \left\{\bar{v}\_{\mathbf{x}}(r)\xi^{t,z;\bar{\mathfrak{a}}}(r) + \frac{1}{2}\xi^{t,z;\bar{\mathfrak{a}}}(r)\,^\top\bar{v}\_{\mathbf{x}\mathbf{x}}(r)\xi^{t,z;\bar{\mathfrak{a}}}(r)\right\} d\mathcal{W}(r) \\ \qquad + \int\_{\mathbb{E}} \left\{\bar{c}\_{\mathbf{x}}(r,e)\xi^{t,z;\mathfrak{a}}(r-) + \frac{1}{2}\xi^{t,z;\mathfrak{a}}(r-)\,^\top\bar{c}\_{\mathbf{x}\mathbf{x}}(r,e)\xi^{t,z;\mathfrak{a}}(r-)\right\} \mathcal{N}(dedr) \\ \qquad + \varepsilon\_{\mathsf{z}4}(r)dr + \varepsilon\_{\mathsf{z}5}(r)dW(r) + \int\_{\mathbb{E}}\varepsilon\_{\mathsf{z}6}(r,e)\bar{\mathcal{N}}(dedr), \quad r \in [t,T], \\ \xi^{t,z;\mathfrak{a}}(t) = z - x^{s,y;\mathfrak{a}}(t), \end{cases} \tag{40}$$

where

$$\begin{cases} \begin{aligned} \varepsilon\_{z4}(r) &:= \int\_{0}^{1} (1-\theta)\xi^{t,z;\overline{\mu}}(r)^{\top} \left[b\_{\text{xx}}(r,\mathfrak{x}^{\sharp,y;\overline{\mu}}(r)+\theta\xi^{t,z;\overline{\mu}}(r),\mathfrak{u}(r))-\bar{b}\_{\text{xx}}(r)\right]\xi^{t,z;\overline{\mu}}(r)d\theta\_{\text{s}} \\ \varepsilon\_{z5}(r) &:= \int\_{0}^{1} (1-\theta)\xi^{t,z;\mu}(r)^{\top} \left[\sigma\_{\text{xx}}(r,\mathfrak{x}^{\sharp,y;\overline{\mu}}(r)+\theta\xi^{t,z;\overline{\mu}}(r),\mathfrak{u}(r))-\bar{\sigma}\_{\text{xx}}(r)\right]\xi^{t,z;\overline{\mu}}(r)d\theta\_{\text{s}} \\ \varepsilon\_{z6}(r,\cdot) &:= \int\_{0}^{1} (1-\theta)\xi^{t,z;\overline{\mu}}(r-)^{\top} \left[c\_{\text{xx}}(r,\mathfrak{x}^{\sharp,y;\overline{\mu}}(r-)+\theta\xi^{t,z;\overline{\mu}}(r-),\mathfrak{u}(r),\cdot)-\bar{\sigma}\_{\text{xx}}(r,\cdot)\right]\xi^{t,z;\overline{\mu}}(r-)d\theta\_{\text{s}} \end{aligned} \end{cases}$$

We are going to show that, there exists a deterministic continuous and increasing function *<sup>δ</sup>* : [0, <sup>∞</sup>) <sup>→</sup> [0, <sup>∞</sup>), independent of *<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***n*, with *<sup>δ</sup>*(*r*) *<sup>r</sup>* → 0 as *r* → 0, such that

$$\begin{cases} \mathbb{E}\left[\int\_{t}^{T} |\varepsilon\_{z1}(r)|^{2} dr |\mathcal{F}\_{t}^{s}| \right](\omega) \leq \delta(|z-\bar{\mathbf{x}}^{s,y;\boldsymbol{\mathcal{R}}}(t,\omega)|^{2}), & \mathbf{P}\text{-a.s.}\omega, \\\mathbb{E}\left[\int\_{t}^{T} |\varepsilon\_{z2}(r)|^{2} dr |\mathcal{F}\_{t}^{s}| \right](\omega) \leq \delta(|z-\bar{\mathbf{x}}^{s,y;\boldsymbol{\mathcal{R}}}(t,\omega)|^{2}), & \mathbf{P}\text{-a.s.}\omega, \\\mathbb{E}\left[\int\_{t}^{T} ||\varepsilon\_{z3}(r,\cdot)||\_{\mathcal{L}^{2}}^{2} dr |\mathcal{F}\_{t}^{s}| \right](\omega) \leq \delta(|z-\bar{\mathbf{x}}^{s,y;\boldsymbol{\mathcal{R}}}(t,\omega)|^{2}), & \mathbf{P}\text{-a.s.}\omega. \end{cases} \tag{41}$$

and

$$\begin{cases} \mathbb{E}\left[\int\_{t}^{T} |\varepsilon\_{z4}(r)|^{2} dr \big| \mathcal{F}\_{t}^{s} \right](\omega) \leq \delta(|z - \bar{\mathbf{x}}^{s,y;\mathcal{R}}(t,\omega)|^{4}), & \mathsf{P-a.s.}\omega, \\\mathbb{E}\left[\int\_{t}^{T} |\varepsilon\_{z5}(r)|^{2} dr \big| \mathcal{F}\_{t}^{s} \right](\omega) \leq \delta(|z - \bar{\mathbf{x}}^{s,y;\mathcal{R}}(t,\omega)|^{4}), & \mathsf{P-a.s.}\omega, \\\mathbb{E}\left[\int\_{t}^{T} ||\varepsilon\_{z6}(r, \cdot)||\_{\mathcal{L}^{2}}^{2} dr \big| \mathcal{F}\_{t}^{s} \right](\omega) \leq \delta(|z - \bar{\mathbf{x}}^{s,y;\mathcal{R}}(t,\omega)|^{4}), & \mathsf{P-a.s.}\omega. \end{cases} \tag{42}$$

#### 14 Will-be-set-by-IN-TECH 132 Stochastic Modeling and Control Stochastic Control for Jump Diffusions <sup>7</sup> <sup>15</sup>

We start to prove (41). To this end, let us fixed an *ω* ∈ Ω such that (38) holds. Then, by setting *bx*(*r*, *θ*) := *bx*(*r*, *x*¯*s*,*y*;*u*¯ (*r*) + *θξt*,*z*;*u*¯ (*r*), *u*¯(*r*)) and in virtue of **(H3)**, we have

**E** � � *<sup>T</sup> t*

> ≤ � *T t* **E** � � <sup>1</sup> 0

≤ � *T t*

≤ *C* � *T t*

*<sup>δ</sup>*(·) independent of *<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***n*.

**E** � � *T t* � ¯

− � *T t* �

<sup>=</sup> �−*p*(*t*), *<sup>ξ</sup>t*,*z*;*u*¯

<sup>+</sup> �*q*(*r*), *<sup>ξ</sup>t*,*z*;*u*¯

⎧

*d*Φ*t*,*z*;*u*¯

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Φ*t*,*z*;*u*¯

Applying Itô's formula to �*ξt*,*z*;*u*¯

*fx*(*r*), *ξt*,*z*;*u*¯

� **E** � *<sup>ω</sup>*¯(|*ξt*,*z*;*u*¯

> � **E** � *<sup>ω</sup>*¯(|*ξt*,*z*;*u*¯

large constant *<sup>C</sup>* <sup>&</sup>gt; 0 and define *<sup>δ</sup>*(*r*) <sup>≡</sup> *Cr*�

(*t*)� − **E**

On the other hand, apply Itô's formula to Φ*t*,*z*;*u*¯

(*r*) = � ¯ *bx*(*r*)Φ*t*,*z*;*u*¯


<sup>L</sup><sup>2</sup> *dr*� �F*s t* � (*ω*)


(*r*−)| 4) � �F*s t* � (*ω*) �1 <sup>2</sup> � **E** � <sup>|</sup>*ξt*,*z*;*u*¯

(*r*)| 4) � �F*s t* � (*ω*) �1 2

(*r*)�*dr* + �*hx*(*x*¯

�1 2 � *T t* �

�*p*(*r*),*εz*4(*r*)� + �*q*(*r*),*εz*5(*r*)� +

+ � **E**

+ �

+ � **E** �

(*t*) = *ξt*,*z*;*u*¯

+ Φ*t*,*z*;*u*¯

(*t*)*ξt*,*z*;*u*¯

(*r*)�*σ*¯*xx*(*r*)*ξt*,*z*;*u*¯

*s*,*y*;*u*¯

�*p*(*r*), *<sup>ξ</sup>t*,*z*;*u*¯

(*r*)� + � **E** <sup>L</sup><sup>2</sup> *<sup>d</sup><sup>θ</sup>* · |*ξt*,*z*;*u*¯

*dr* · |*z* − *x*¯

�

(*T*)� � �F*s t* �

*bxx*(*r*)*ξt*,*z*;*u*¯

�*γ*(*r*,*e*),*εz*6(*r*,*e*)�*π*(*de*)

(*r*)*c*¯*x*(*r*,*e*)�*π*(*de*) + *εz*7(*r*)

(*r*) + Φ*z*(*r*)*σ*¯*x*(*r*)� + *εz*8(*r*)

(*r*) := *ξt*,*z*;*u*¯

(*r*)¯

(*r*)�

(*r*)�*c*¯*xx*(*r*,*e*)*ξt*,*z*;*u*¯

(*r*)*ξt*,*z*;*u*¯

*bx*(*r*)� + *σ*¯*x*(*r*)Φ*t*,*z*;*u*¯

(*r*−)*c*¯*x*(*r*,*e*)� <sup>+</sup> *<sup>c</sup>*¯*x*(*r*,*e*)Φ*t*,*z*;*u*¯

�

� *dr* � � �F*s t* �

This yields the third inequality in (42) for an obvious *δ*(·) as above. Finally, we can select the largest *δ*(·) obtained in the above six calculations. For example, we can choose an enough

(*T*)), *ξt*,*z*;*u*¯

(*r*)� ¯

�*γ*(*r*,*e*), *<sup>ξ</sup>t*,*z*;*u*¯

� **E**

(*r*) + Φ*t*,*z*;*u*¯

*c*¯*x*(*r*,*e*)Φ*t*,*z*;*u*¯

*c*¯*x*(*r*,*e*)Φ*t*,*z*;*u*¯

(*t*)�,

(*r*−)*c*¯*x*(*r*,*e*)� + *εz*9(*r*,*e*)

*σ*¯*x*(*r*)Φ*t*,*z*;*u*¯

*<sup>r</sup>* <sup>∨</sup> �*ω*¯(*r*)

(·), *p*(·)�, noting (18) and (40), we have

(*r*−)| 4� �F*s t* � (*ω*)*dr*

(*r*−)| 8� �F*s t* � (*ω*) �1 2 *dr*

*s*,*y*;*u*¯

(*t*, *ω*)| 4.

,*r* ≥ 0. Then (41), (42) follows with a

Stochastic Control for Jump Diff usions 133

(*r*)�*π*(*de*)

(*r*)�, noting (39), we get

(*r*)*σ*¯*x*(*r*)�

(*r*−)

� *dr*

� *dW*(*r*)

*<sup>N</sup>*˜ (*dedr*), *<sup>r</sup>* <sup>∈</sup> [*t*, *<sup>T</sup>*],

� *dr* (45)

(46)

, **P**-*a*.*s*.

$$\begin{split} &\mathbb{E}\left[\int\_{t}^{T} |\varepsilon\_{1}(r)|^{2} dr \, \middle| \, \mathcal{F}\_{t}^{\mathrm{s}} \right](\omega) \\ &\leq \int\_{t}^{T} \mathbb{E}\left\{\int\_{0}^{1} |b\_{\mathrm{x}}(r,\theta) - \bar{b}\_{\mathrm{x}}(r)|^{2} d\theta \, \cdot |\xi^{t,\mathrm{z};\mathrm{fl}}(r)|^{2} \, \middle| \, \mathcal{F}\_{t}^{\mathrm{s}} \right\}(\omega) dr \\ &\leq \mathbb{C} \int\_{t}^{T} \mathbb{E}\left[|\xi^{t,\mathrm{z};\mathrm{fl}}(r)|^{4} |\mathcal{F}\_{t}^{\mathrm{s}}| \, \middle| \, \omega \right] dr \leq \mathbb{C} |z - \bar{\mathbf{x}}^{\mathrm{s},\mathrm{y};\mathrm{fl}}(t,\omega)|^{4}. \end{split}$$

Thus, the first inequality in (41) follows if we choose *<sup>δ</sup>*(*r*) <sup>≡</sup> *Cr*2,*<sup>r</sup>* <sup>≥</sup> 0. The second inequality in (41) can be proved similarly. Setting *cx*(*r*, *<sup>θ</sup>*, ·) :<sup>=</sup> *cx*(*r*, *<sup>x</sup>*¯*s*,*y*;*u*¯ (*r*−) + *θξt*,*z*;*u*¯ (*r*−), *u*¯(*r*), ·) and using **(H3)**, we have

$$\begin{split} &\mathbb{E}\left[\int\_{t}^{T}||\boldsymbol{\varepsilon}\_{z;3}(\boldsymbol{r},\boldsymbol{\gamma})||\_{\mathcal{L}^{2}}^{2} d\boldsymbol{r} |\mathcal{F}\_{t}^{s}\right](\boldsymbol{\omega}) \\ &\leq \int\_{t}^{T}\mathbb{E}\left\{\int\_{0}^{1}||\boldsymbol{c}\_{x}(\boldsymbol{r},\boldsymbol{\theta},\boldsymbol{\cdot})-\boldsymbol{\bar{c}}\_{x}(\boldsymbol{r},\boldsymbol{\cdot})||\_{\mathcal{L}^{2}}^{2} d\boldsymbol{\theta}\cdot\big|\boldsymbol{\xi}^{t,z;3}(\boldsymbol{r}-\boldsymbol{\})^{2}\big|\mathcal{F}\_{t}^{s}\right\}(\boldsymbol{\omega})d\boldsymbol{r} \\ &\leq C\int\_{t}^{T}\mathbb{E}\left[|\boldsymbol{\xi}^{t,z;3}(\boldsymbol{r}-\boldsymbol{\})^{4}\big|\mathcal{F}\_{t}^{s}\right](\boldsymbol{\omega})d\boldsymbol{r} \\ &= C\int\_{t}^{T}\mathbb{E}\left[|\boldsymbol{\xi}^{t,z;3}(\boldsymbol{r})|^{4}\big|\mathcal{F}\_{t}^{s}\right](\boldsymbol{\omega})d\boldsymbol{r} \leq C|\boldsymbol{z}-\boldsymbol{\bar{x}}^{\boldsymbol{\xi};\boldsymbol{\mathcal{H}}}(\boldsymbol{t},\boldsymbol{\omega})|^{4}. \end{split} \tag{43}$$

The equality in (43) holds because the discontinuous points of *ξt*,*z*;*u*¯ (·) are at most countable. Thus, the third inequality in (41) follows for an obvious *δ*(·) as above.

We continue to prove (42). Let *bxx*(*r*, *θ*) := *bxx*(*r*, *x*¯*s*,*y*;*u*¯ (*r*) + *θξt*,*z*;*u*¯ (*r*), *u*¯(*r*)). Using **(H3)**, we can show that

$$\begin{split} & \mathbb{E} \left[ \int\_{t}^{T} |\varepsilon\_{4z}(r)|^{2} dr \Big| \mathcal{F}\_{t}^{s} \right](\omega) \\ & \leq \int\_{t}^{T} \mathbb{E} \left[ \int\_{0}^{1} |b\_{\text{xx}}(r, \theta) - \bar{b}\_{\text{xx}}(r)|^{2} d\theta \cdot |\xi^{t, z; \mathfrak{a}}(r)|^{4} \Big| \mathcal{F}\_{t}^{s} \right](\omega) dr \\ & \leq \int\_{t}^{T} \left\{ \mathbb{E} \left[ \bar{\omega} (|\xi^{t, z; \mathfrak{a}}(r)|^{4}) |\mathcal{F}\_{t}^{s} \right](\omega) \right\}^{\frac{1}{2}} \left\{ \mathbb{E} \left[ |\xi^{t, z; \mathfrak{a}}(r)|^{8} |\mathcal{F}\_{t}^{s} \right](\omega) \right\}^{\frac{1}{2}} dr \\ & \leq C \int\_{t}^{T} \left\{ \mathbb{E} \left[ \bar{\omega} (|\xi^{t, z; \mathfrak{a}}(r)|^{4}) |\mathcal{F}\_{t}^{s} \right](\omega) \right\}^{\frac{1}{2}} dr \cdot |z - \bar{\pi}^{s, \mathfrak{a}}(t, \omega)|^{4} . \end{split} \tag{44}$$

This yields the first inequality in (42) if we choose *<sup>δ</sup>*(*r*) <sup>≡</sup> *Crω*¯(*r*),*<sup>r</sup>* <sup>≥</sup> 0. Noting that the modulus of continuity *ω*¯(·) is defined in **(H3)**. The second inequality in (42) can be proved similarly. Setting *cxx*(*r*, *<sup>θ</sup>*, ·) :<sup>=</sup> *cxx*(*r*, *<sup>x</sup>*¯*s*,*y*;*u*¯ (*r*−) + *θξt*,*z*;*u*¯ (*r*−), *u*¯(*r*), ·) and by virtue of **(H3)**, noting the remark following (43), we show that

$$\begin{split} &\mathbb{E}\left[\int\_{t}^{T}||e\_{5}(r,\cdot)||\_{\mathcal{L}^{2}}^{2}dr|\mathcal{F}\_{t}^{\boldsymbol{s}}\right](\omega) \\ &\leq\int\_{t}^{T}\mathbb{E}\left[\int\_{0}^{1}||e\_{\mathrm{xx}}(r,\theta,\cdot)-\bar{e}\_{\mathrm{xx}}(r,\cdot)||\_{\mathcal{L}^{2}}^{2}d\theta\cdot|\tilde{\xi}^{t,\boldsymbol{z};\boldsymbol{\theta}}(r-)|^{4}\big|\mathcal{F}\_{t}^{\boldsymbol{s}}\right](\omega)dr \\ &\leq\int\_{t}^{T}\left\{\mathbb{E}\left[\bar{\omega}\big(|\xi^{t,\boldsymbol{z};\bar{\boldsymbol{u}}}(r-)|^{4}\big|\mathcal{F}\_{t}^{\boldsymbol{s}}\right)(\omega)\right\}^{\frac{1}{2}}\left\{\mathbb{E}\left[|\xi^{t,\boldsymbol{z};\bar{\boldsymbol{u}}}(r-)|^{8}\big|\mathcal{F}\_{t}^{\boldsymbol{s}}\right](\omega)\right\}^{\frac{1}{2}}dr \\ &\leq\mathbb{C}\int\_{t}^{T}\left\{\mathbb{E}\left[\bar{\omega}\big(|\xi^{t,\boldsymbol{z};\boldsymbol{\theta}}(r)|^{4}\big|\mathcal{F}\_{t}^{\boldsymbol{s}}\right)(\omega)\right\}^{\frac{1}{2}}dr\cdot|\boldsymbol{z}-\bar{\mathbf{x}}^{\boldsymbol{s},\boldsymbol{y};\boldsymbol{\theta}}(\boldsymbol{t},\boldsymbol{\omega})|^{4}.\end{split}$$

This yields the third inequality in (42) for an obvious *δ*(·) as above. Finally, we can select the largest *δ*(·) obtained in the above six calculations. For example, we can choose an enough large constant *<sup>C</sup>* <sup>&</sup>gt; 0 and define *<sup>δ</sup>*(*r*) <sup>≡</sup> *Cr*� *<sup>r</sup>* <sup>∨</sup> �*ω*¯(*r*) � ,*r* ≥ 0. Then (41), (42) follows with a *<sup>δ</sup>*(·) independent of *<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***n*.

Applying Itô's formula to �*ξt*,*z*;*u*¯ (·), *p*(·)�, noting (18) and (40), we have

14 Will-be-set-by-IN-TECH

We start to prove (41). To this end, let us fixed an *ω* ∈ Ω such that (38) holds. Then, by setting

*bx*(*r*)|

Thus, the first inequality in (41) follows if we choose *<sup>δ</sup>*(*r*) <sup>≡</sup> *Cr*2,*<sup>r</sup>* <sup>≥</sup> 0. The second inequality

(*r*), *u*¯(*r*)) and in virtue of **(H3)**, we have

<sup>2</sup>*d<sup>θ</sup>* · |*ξt*,*z*;*u*¯

(*ω*)*dr* ≤ *C*|*z* − *x*¯

<sup>L</sup><sup>2</sup> *<sup>d</sup><sup>θ</sup>* · |*ξt*,*z*;*u*¯

(*ω*)*dr* ≤ *C*|*z* − *x*¯

<sup>2</sup>*d<sup>θ</sup>* · |*ξt*,*z*;*u*¯

(*r*)| 2 F*s t* (*ω*)*dr*

*s*,*y*;*u*¯

(*r*−)| 2 F*s t* (*ω*)*dr*

(*t*, *ω*)| 4.

*s*,*y*;*u*¯

(*r*) + *θξt*,*z*;*u*¯

(*r*)| 4 F*s t* (*ω*)*dr*

*dr* · |*z* − *x*¯

*s*,*y*;*u*¯

(*t*, *ω*)| 4.

(*r*−), *u*¯(*r*), ·) and by virtue of **(H3)**,

(*t*, *ω*)| 4.

(*r*−) + *θξt*,*z*;*u*¯

(*r*−), *u*¯(*r*), ·) and

(·) are at most countable.

(*r*), *u*¯(*r*)). Using **(H3)**, we

(43)

(44)

*bx*(*r*, *θ*) := *bx*(*r*, *x*¯*s*,*y*;*u*¯

using **(H3)**, we have

can show that

**E** *<sup>T</sup> t*

> ≤ *T t* **E** 1 0

≤ *C*

= *C T t* **E** <sup>|</sup>*ξt*,*z*;*u*¯ (*r*)| 4 F*s t* 

**E** *<sup>T</sup> t*

> ≤ *T t* **E** <sup>1</sup> 0

≤ *T t*

≤ *C T t*

 *T t* **E** <sup>|</sup>*ξt*,*z*;*u*¯

(*r*) + *θξt*,*z*;*u*¯


in (41) can be proved similarly. Setting *cx*(*r*, *<sup>θ</sup>*, ·) :<sup>=</sup> *cx*(*r*, *<sup>x</sup>*¯*s*,*y*;*u*¯

<sup>L</sup><sup>2</sup> *dr* F*s t* (*ω*)

(*r*−)| 4 F*s t* (*ω*)*dr*

The equality in (43) holds because the discontinuous points of *ξt*,*z*;*u*¯

We continue to prove (42). Let *bxx*(*r*, *θ*) := *bxx*(*r*, *x*¯*s*,*y*;*u*¯

<sup>2</sup>*dr* F*s t* (*ω*)


 **E** *<sup>ω</sup>*¯(|*ξt*,*z*;*u*¯

similarly. Setting *cxx*(*r*, *<sup>θ</sup>*, ·) :<sup>=</sup> *cxx*(*r*, *<sup>x</sup>*¯*s*,*y*;*u*¯

noting the remark following (43), we show that

 **E** *<sup>ω</sup>*¯(|*ξt*,*z*;*u*¯

Thus, the third inequality in (41) follows for an obvious *δ*(·) as above.

<sup>|</sup>*bxx*(*r*, *<sup>θ</sup>*) <sup>−</sup> ¯

(*r*)| 4) F*s t* (*ω*) 1 <sup>2</sup> **E** <sup>|</sup>*ξt*,*z*;*u*¯ (*r*)| 8 F*s t* (*ω*) 1 2 *dr*

(*r*)| 4) F*s t* (*ω*) 1 2

*bxx*(*r*)|

This yields the first inequality in (42) if we choose *<sup>δ</sup>*(*r*) <sup>≡</sup> *Crω*¯(*r*),*<sup>r</sup>* <sup>≥</sup> 0. Noting that the modulus of continuity *ω*¯(·) is defined in **(H3)**. The second inequality in (42) can be proved

(*r*−) + *θξt*,*z*;*u*¯


<sup>2</sup>*dr* F*s t* (*ω*)

<sup>|</sup>*bx*(*r*, *<sup>θ</sup>*) <sup>−</sup> ¯

**E** *<sup>T</sup> t*

> ≤ *T t* **E** 1 0

≤ *C*

 *T t* **E** <sup>|</sup>*ξt*,*z*;*u*¯ (*r*)| 4 F*s t* 


$$\begin{split} & \mathbb{E} \left\{ \int\_{t}^{T} \langle \tilde{f}\_{\mathbf{x}}(r), \xi^{t,z;\mathcal{U}}(r) \rangle dr + \langle h\_{\mathbf{x}}(\tilde{\mathbf{x}}^{s,y;\mathcal{U}}(T)), \xi^{t,z;\mathcal{U}}(T) \rangle \Big| \, \mathcal{F}\_{t}^{s} \right\} \\ &= \langle -p(t), \xi^{t,z;\mathcal{U}}(t) \rangle - \mathbb{E} \left\{ \frac{1}{2} \int\_{t}^{T} \left[ \langle p(r), \xi^{t,z;\mathcal{U}}(r) \rangle^{\top} \bar{b}\_{\mathbf{x}\mathbf{x}}(r) \xi^{t,z;\mathcal{U}}(r) \right] \right. \\ & \left. + \langle q(r), \xi^{t,z;\mathcal{U}}(r) \rangle^{\top} \bar{o}\_{\mathbf{x}\mathbf{x}}(r) \xi^{t,z;\mathcal{U}}(r) \rangle + \int\_{\mathcal{E}} \langle \gamma(r,e), \xi^{t,z;\mathcal{U}}(r) \upharpoonright^{\top} \bar{c}\_{\mathbf{x}\mathbf{x}}(r,e) \xi^{t,z;\mathcal{U}}(r) \rangle \, \pi(de) \Big] dr \\ & - \int\_{t}^{T} \left[ \langle p(r), \varepsilon\_{z4}(r) \rangle + \langle q(r), \varepsilon\_{z5}(r) \rangle + \int\_{\mathcal{E}} \langle \gamma(r,e), \varepsilon\_{z6}(r,e) \rangle \, \pi(de) \right] dr \bigg| \mathcal{F}\_{t}^{s} \right\}, \quad \mathbf{P} \text{-a.s.} \end{split} \tag{45}$$

On the other hand, apply Itô's formula to Φ*t*,*z*;*u*¯ (*r*) := *ξt*,*z*;*u*¯ (*r*)*ξt*,*z*;*u*¯ (*r*)�, noting (39), we get

⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *d*Φ*t*,*z*;*u*¯ (*r*) = � ¯ *bx*(*r*)Φ*t*,*z*;*u*¯ (*r*) + Φ*t*,*z*;*u*¯ (*r*)¯ *bx*(*r*)� + *σ*¯*x*(*r*)Φ*t*,*z*;*u*¯ (*r*)*σ*¯*x*(*r*)� + � **E** *c*¯*x*(*r*,*e*)Φ*t*,*z*;*u*¯ (*r*)*c*¯*x*(*r*,*e*)�*π*(*de*) + *εz*7(*r*) � *dr* + � *σ*¯*x*(*r*)Φ*t*,*z*;*u*¯ (*r*) + Φ*z*(*r*)*σ*¯*x*(*r*)� + *εz*8(*r*) � *dW*(*r*) + � **E** � *c*¯*x*(*r*,*e*)Φ*t*,*z*;*u*¯ (*r*−)*c*¯*x*(*r*,*e*)� <sup>+</sup> *<sup>c</sup>*¯*x*(*r*,*e*)Φ*t*,*z*;*u*¯ (*r*−) + Φ*t*,*z*;*u*¯ (*r*−)*c*¯*x*(*r*,*e*)� + *εz*9(*r*,*e*) � *<sup>N</sup>*˜ (*dedr*), *<sup>r</sup>* <sup>∈</sup> [*t*, *<sup>T</sup>*], Φ*t*,*z*;*u*¯ (*t*) = *ξt*,*z*;*u*¯ (*t*)*ξt*,*z*;*u*¯ (*t*)�, (46)

#### 16 Will-be-set-by-IN-TECH 134 Stochastic Modeling and Control Stochastic Control for Jump Diffusions <sup>8</sup> <sup>17</sup>

where

⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *εz*7(*r*) := *εz*1(*r*)*ξt*,*z*;*u*¯ (*r*)� + *ξt*,*z*;*u*¯ (*r*)*εz*1(*r*)� + *σ*¯*x*(*r*)*ξt*,*z*;*u*¯ (*r*)*εz*2(*r*)� + *εz*2(*r*)*ξt*,*z*;*u*¯ (*r*)�*σ*¯*x*(*r*)� + *εz*2(*r*)*εz*2(*r*)� + � **E** � *c*¯*x*(*r*,*e*)*ξt*,*z*;*u*¯ (*r*)*εz*3(*r*,*e*)� + *εz*3(*r*,*e*)*ξt*,*z*;*u*¯ (*r*)�*c*¯*x*(*r*,*e*)� <sup>+</sup> *<sup>ε</sup>z*3(*r*,*e*)*εz*3(*r*,*e*)�� *π*(*de*), *εz*8(*r*) := *εz*2(*r*)*ξt*,*z*;*u*¯ (*r*)� + *ξt*,*z*;*u*¯ (*r*)*εz*2(*r*)�, *<sup>ε</sup>z*9(*r*, ·) :<sup>=</sup> *<sup>c</sup>*¯*x*(*r*, ·)*ξt*,*z*;*u*¯ (*r*−)*εz*3(*r*, ·)� <sup>+</sup> *<sup>ε</sup>z*3(*r*, ·)*ξt*,*z*;*u*¯ (*r*−)�*c*¯*x*(*r*, ·)� + *ξt*,*z*;*u*¯ (*r*−)*εz*3(*r*, ·)� <sup>+</sup> *<sup>ε</sup>z*3(*r*, ·)*ξt*,*z*;*u*¯ (*r*−)� + *εz*3(*r*, ·)*εz*3(*r*, ·)�.

By virtue of (45) and (47), we have

(*T*)�*hxx*(*x*¯

*s*,*y*;*u*¯

Note that the term *<sup>o</sup>*(|*<sup>z</sup>* <sup>−</sup> *<sup>x</sup>*¯*s*,*y*;*u*¯

= −�*p*(*t*, *ω*) + *p*, *z* − *x*¯

*s*,*y*;*u*¯

+ *o*(|*z* − *x*¯

Then, it is necessary that

*s*,*y*;*u*¯

(*t*, *ω*0))

*s*,*y*;*u*¯

*s*,*y*;*u*¯

(*t*, *ω*0)|

*<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***n*, which by definition (30) proves

*<sup>x</sup> V*(*t*, *x*¯*s*,*y*;*u*¯

0 ≤ *V*(*t*, *z*) − *V*(*t*, *x*¯(*t*, *ω*)) − �*p*, *z* − *x*¯

(*t*, *ω*)| 2).

(*t*, *<sup>ω</sup>*0)� − <sup>1</sup>

(*t*, *<sup>ω</sup>*0)� − <sup>1</sup>

2).

(−*p*(*t*), <sup>−</sup>*P*(*t*)) ∈ P2,<sup>+</sup>

*s*,*y*;*u*¯

Thus, (35) holds. (36) is immediate. The proof is complete.

*s*,*y*;*u*¯

may happen in some cases, as shown in the following example.

*s*,*y*;*u*¯

(*<sup>n</sup>* <sup>=</sup> <sup>1</sup>): *dxs*,*y*;*u*(*t*) = *<sup>u</sup>*(*t*)*dt* <sup>+</sup> *<sup>δ</sup>u*(*t*)*N*˜ (*dt*), *<sup>t</sup>* <sup>∈</sup> [*s*, *<sup>T</sup>*],

2 **E** *T s*

*xs*,*y*;*u*(*s*) = *y*,

*<sup>J</sup>*(*s*, *<sup>y</sup>*; *<sup>u</sup>*(·)) = <sup>1</sup>

*Vx*(*t*, *<sup>x</sup>*¯

*Vxx*(*t*, *x*¯

2 **E** *<sup>T</sup> t*

2 *ξt*,*z*;*u*¯

(*t*, *<sup>ω</sup>*0)� − <sup>1</sup>

(*T*) F*s t* 

> 2 (*z* − *x*¯

(*t*, *ω*0)|

(*T*))*ξt*,*z*;*u*¯

*ξt*,*z*;*u*¯

(*r*)�*H*¯ *xx*(*r*)*ξt*,*z*;*u*¯

(*ω*0) + *o*(|*z* − *x*¯

(*t*, *ω*0)�*P*(*t*, *ω*0)*ξt*,*z*;*u*¯

*s*,*y*;*u*¯

it is independent of *z*. Therefore, by the continuity of *V*(*t*, ·), we see that (49) holds for all

Then (34) holds. Let us now show (35). Fix an *<sup>ω</sup>* <sup>∈</sup> <sup>Ω</sup> such that (49) holds for any *<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***n*. For

(*t*)), by definition (30) and (49) we have

*p* = −*p*(*t*), *P* ≤ −*P*(*t*), **P**-*a*.*s*.

**Remark 3.2** It is interesting to note that if *<sup>V</sup>* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*), then (34)∼(35) reduce to

**Example 3.1** Consider the following linear stochastic control system with Poisson jumps

for some *<sup>δ</sup>* �<sup>=</sup> 0. Here *<sup>N</sup>* is a Poisson process with the intensity *<sup>λ</sup>dt* and *<sup>N</sup>*˜ (*dt*) :<sup>=</sup> *<sup>N</sup>*(*dt*) <sup>−</sup> *λdt*(*λ* > 0) is the compensated martingale measure. The quadratic cost functional is taken as

> *<sup>λ</sup>* <sup>−</sup> <sup>1</sup> <sup>−</sup> *<sup>λ</sup> <sup>λ</sup> <sup>e</sup> t*−*T δ*2

1

(*t*)) = −*p*(*t*),

We point out that in our jump diffusion setting the strict inequality *Vxx*(*t*, *x*¯*s*,*y*;*u*¯

(*t*, *<sup>ω</sup>*)� − <sup>1</sup>

*s*,*y*;*u*¯

2 (*z* − *x*¯

*s*,*y*;*u*¯

*<sup>x</sup> V*(*t*, *x*¯

*s*,*y*;*u*¯

2 (*z* − *x*¯

(*t*, *<sup>ω</sup>*)� − <sup>1</sup>

(*r*)*dr*

(*t*, *ω*0)| 2)

(*t*, *ω*0) + *o*(|*z* − *x*¯

*s*,*y*;*u*¯

(*t*, *ω*))�*P*(*z* − *x*¯

*s*,*y*;*u*¯

Stochastic Control for Jump Diff usions 135

(*t*, *ω*0))

(*t*, *ω*0)| 2)

(*t*, *ω*0)|

*s*,*y*;*u*¯

*s*,*y*;*u*¯

(*t*, *ω*))

(*t*, *ω*))

(*t*)) < −*P*(*t*)

(49)

2, and

*s*,*y*;*u*¯

(*t*, *ω*0))�*P*(*t*, *ω*0)(*z* − *x*¯

<sup>2</sup>) depends only on the size of <sup>|</sup>*<sup>z</sup>* <sup>−</sup> *<sup>x</sup>*¯*s*,*y*;*u*¯

(*t*)), ∀*t* ∈ [*s*, *T*], **P**-*a*.*s*.

*s*,*y*;*u*¯

(*t*, *ω*))�(*P*(*t*, *ω*) + *P*)(*z* − *x*¯

(*t*)) ≤ −*P*(*t*), <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [*s*, *<sup>T</sup>*], **<sup>P</sup>**-*a*.*s*. (50)

*xs*,*y*;*u*(*t*)2*dt*.

*V*(*t*, *z*) − *V*(*t*, *x*¯

≤ −�*p*(*t*, *<sup>ω</sup>*0), *<sup>ξ</sup>t*,*z*;*u*¯

<sup>=</sup> − �*p*(*t*, *<sup>ω</sup>*0), *<sup>ξ</sup>t*,*z*;*u*¯

= − �*p*(*t*, *ω*0), *z* − *x*¯

+ *o*(|*z* − *x*¯

any (*p*, *<sup>P</sup>*) ∈ P2,<sup>−</sup>

+ *ξt*,*z*;*u*¯

Once more applying Itô's formula to Φ*t*,*z*;*u*¯ (·)�*P*(·), noting (19) and (46), we get

$$\begin{split} & - \mathbb{E} \left\{ \int\_{t}^{\top} \xi^{t, \sharp \underline{a}}(r)^{\top} \bar{H}\_{\scriptscriptstyle\rm XY}(r) \xi^{t, \sharp \underline{a}}(r) dr + \xi^{t, \sharp \underline{a}}(T)^{\top} h\_{\scriptscriptstyle\rm XY}(\bar{x}^{\prime \sharp \underline{a}}(T)) \xi^{t, \sharp \underline{a}}(T) |\mathcal{F}\_{t}^{\rm S} \right\} \\ &= - \xi^{t, \sharp \underline{a}}(t)^{\top} P(t) \xi^{t, \sharp \underline{a}}(t) \\ & - \mathbb{E} \left\{ \int\_{t}^{\top} \text{tr} \left\{ P(r) \varepsilon\_{\mathcal{Z}}(r) + Q(r) \varepsilon\_{\mathcal{Z} \ $}(r) + \int\_{\mathbb{E}} R(r, e) \varepsilon\_{\mathcal{Z} \$ }(r, e) \pi(de) \right\} dr |\mathcal{F}\_{t}^{\rm S} \right\}\_{t} \quad \text{P-a.s.} \end{split} (47)$$

Let us call a *<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup> rational* if all its coordinate are rational. Since the set of all rational *<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* is countable, we may find a *common* subset Ω<sup>0</sup> ⊆ Ω with *P*(Ω0) = 1 such that for any *ω*<sup>0</sup> ∈ Ω0,

$$\begin{cases} V(t, \mathfrak{x}^{\mathfrak{s}, \mathfrak{z}; \widetilde{\mu}}(t, \omega\_0)) = \mathbb{E} \left[ \int\_s^T \widetilde{f}(r) dr + h(\mathfrak{x}^{\mathfrak{s}, \mathfrak{z}; \widetilde{\mu}}(T)) \Big| \mathcal{F}\_t^{\mathfrak{s}} \right](\omega\_0), \\\\ (38), (41), (42), (45), (47) \text{ are satisfied for any rational } z\_\prime \text{ and } \mathfrak{z}(\cdot)|\_{[t, T]} \in \mathcal{U}[t, T]. \end{cases}$$

Let *<sup>ω</sup>*<sup>0</sup> <sup>∈</sup> <sup>Ω</sup><sup>0</sup> be fixed, then for any rational *<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***n*, noting (41) and (42), we have

$$\begin{split} &V(t,\boldsymbol{z}) - V(t,\boldsymbol{\bar{x}}^{\boldsymbol{y},\boldsymbol{y};\boldsymbol{\bar{R}}}(t,\boldsymbol{\omega}\_{0})) \\ &= -\mathbb{E}\Big{{}\Big{{}}\Big{{}}^{T}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}}\Big{{}$$

By virtue of (45) and (47), we have

16 Will-be-set-by-IN-TECH

(*r*)�*σ*¯*x*(*r*)� + *εz*2(*r*)*εz*2(*r*)� +

(*r*)*εz*2(*r*)�,

(*r*−)*εz*3(*r*, ·)� <sup>+</sup> *<sup>ε</sup>z*3(*r*, ·)*ξt*,*z*;*u*¯

(*r*)*dr* + *ξt*,*z*;*u*¯

� **E**

Let us call a *<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup> rational* if all its coordinate are rational. Since the set of all rational *<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***<sup>n</sup>* is countable, we may find a *common* subset Ω<sup>0</sup> ⊆ Ω with *P*(Ω0) = 1 such that for any *ω*<sup>0</sup> ∈ Ω0,

> *s*,*y*;*u*¯ (*T*))� �F*s t* � (*ω*0),

(38),(41),(42),(45),(47) are satisfied for any rational *z*, and *u*¯(·)|[*t*,*T*] ∈ U[*t*, *T*].

*dr* + *h*(*xt*,*z*;*u*¯

(*T*)), *ξt*,*z*;*u*¯

(*r*)*dr* + *ξt*,*z*;*u*¯

*f*(*r*)*dr* + *h*(*x*¯

Let *<sup>ω</sup>*<sup>0</sup> <sup>∈</sup> <sup>Ω</sup><sup>0</sup> be fixed, then for any rational *<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***n*, noting (41) and (42), we have

*f*(*r*) �

*s*,*y*;*u*¯

(*r*−)*εz*3(*r*, ·)� <sup>+</sup> *<sup>ε</sup>z*3(*r*, ·)*ξt*,*z*;*u*¯

(*r*)*εz*1(*r*)� + *σ*¯*x*(*r*)*ξt*,*z*;*u*¯

(*r*)�*c*¯*x*(*r*,*e*)� <sup>+</sup> *<sup>ε</sup>z*3(*r*,*e*)*εz*3(*r*,*e*)��

(*r*)*εz*2(*r*)�

*c*¯*x*(*r*,*e*)*ξt*,*z*;*u*¯

*π*(*de*),

(*r*−)�*c*¯*x*(*r*, ·)�

(·)�*P*(·), noting (19) and (46), we get

*s*,*y*;*u*¯

(*T*)�*hxx*(*x*¯

*R*(*r*,*e*)*εz*9(*r*,*e*)*π*(*de*)

(*T*)) − *h*(*x*¯

(*T*)� � �F*s t* � (*ω*0)

(*T*)�*hxx*(*x*¯

*s*,*y*;*u*¯ (*T*))� �F*s t* � (*ω*0)

*s*,*y*;*u*¯

(*T*))*ξt*,*z*;*u*¯

(*T*) � �F*s t* � (*ω*0)

(*r*−)� + *εz*3(*r*, ·)*εz*3(*r*, ·)�.

(*T*))*ξt*,*z*;*u*¯

� *dr*� �F*s t* �

(*T*) � �F*s t* �

, **P**-*a*.*s*.

(47)

(48)

(*r*)*εz*3(*r*,*e*)�

� **E** �

(*r*)� + *ξt*,*z*;*u*¯

(*r*)� + *ξt*,*z*;*u*¯

where ⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

*εz*7(*r*) := *εz*1(*r*)*ξt*,*z*;*u*¯

*εz*8(*r*) := *εz*2(*r*)*ξt*,*z*;*u*¯

*<sup>ε</sup>z*9(*r*, ·) :<sup>=</sup> *<sup>c</sup>*¯*x*(*r*, ·)*ξt*,*z*;*u*¯

+ *ξt*,*z*;*u*¯

(*r*)�*H*¯ *xx*(*r*)*ξt*,*z*;*u*¯

(*t*)

*P*(*r*)*εz*7(*r*) + *Q*(*r*)*εz*8(*r*) +

� � *<sup>T</sup> s* ¯

Once more applying Itô's formula to Φ*t*,*z*;*u*¯

*ξt*,*z*;*u*¯

(*t*)�*P*(*t*)*ξt*,*z*;*u*¯

(*t*, *ω*0)) = **E**

*s*,*y*;*u*¯

*f*(*r*, *xt*,*z*;*u*¯

*fx*(*r*), *ξt*,*z*;*u*¯

*ξt*,*z*;*u*¯

*s*,*y*;*u*¯

(*r*)� ¯

2).

(*t*, *ω*0)|

(*t*, *ω*0))

(*r*), *<sup>u</sup>*¯(*r*)) <sup>−</sup> ¯

(*r*)� + �*hx*(*x*¯

*fxx*(*r*)*ξt*,*z*;*u*¯

+ *εz*2(*r*)*ξt*,*z*;*u*¯

+ *εz*3(*r*,*e*)*ξt*,*z*;*u*¯

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

− **E**

<sup>=</sup> <sup>−</sup> *<sup>ξ</sup>t*,*z*;*u*¯

− **E**

⎧ ⎪⎪⎨

*V*(*t*, *x*¯ *s*,*y*;*u*¯

*V*(*t*, *z*) − *V*(*t*, *x*¯

� � *<sup>T</sup> t* �

� � *<sup>T</sup> t* � ¯

+ *o*(|*z* − *x*¯

⎪⎪⎩

= **E**

= **E**

+ 1 2 **E** � � *<sup>T</sup> t*

� � *<sup>T</sup> t*

� � *<sup>T</sup> t* tr �

*V*(*t*, *z*) − *V*(*t*, *x*¯ *s*,*y*;*u*¯ (*t*, *ω*0)) ≤ −�*p*(*t*, *<sup>ω</sup>*0), *<sup>ξ</sup>t*,*z*;*u*¯ (*t*, *<sup>ω</sup>*0)� − <sup>1</sup> 2 **E** *<sup>T</sup> t ξt*,*z*;*u*¯ (*r*)�*H*¯ *xx*(*r*)*ξt*,*z*;*u*¯ (*r*)*dr* + *ξt*,*z*;*u*¯ (*T*)�*hxx*(*x*¯ *s*,*y*;*u*¯ (*T*))*ξt*,*z*;*u*¯ (*T*) F*s t* (*ω*0) + *o*(|*z* − *x*¯ *s*,*y*;*u*¯ (*t*, *ω*0)| 2) <sup>=</sup> − �*p*(*t*, *<sup>ω</sup>*0), *<sup>ξ</sup>t*,*z*;*u*¯ (*t*, *<sup>ω</sup>*0)� − <sup>1</sup> 2 *ξt*,*z*;*u*¯ (*t*, *ω*0)�*P*(*t*, *ω*0)*ξt*,*z*;*u*¯ (*t*, *ω*0) + *o*(|*z* − *x*¯ *s*,*y*;*u*¯ (*t*, *ω*0)| 2) = − �*p*(*t*, *ω*0), *z* − *x*¯ *s*,*y*;*u*¯ (*t*, *<sup>ω</sup>*0)� − <sup>1</sup> 2 (*z* − *x*¯ *s*,*y*;*u*¯ (*t*, *ω*0))�*P*(*t*, *ω*0)(*z* − *x*¯ *s*,*y*;*u*¯ (*t*, *ω*0)) + *o*(|*z* − *x*¯ *s*,*y*;*u*¯ (*t*, *ω*0)| 2). (49)

Note that the term *<sup>o</sup>*(|*<sup>z</sup>* <sup>−</sup> *<sup>x</sup>*¯*s*,*y*;*u*¯ (*t*, *ω*0)| <sup>2</sup>) depends only on the size of <sup>|</sup>*<sup>z</sup>* <sup>−</sup> *<sup>x</sup>*¯*s*,*y*;*u*¯ (*t*, *ω*0)| 2, and it is independent of *z*. Therefore, by the continuity of *V*(*t*, ·), we see that (49) holds for all *<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***n*, which by definition (30) proves

$$(-p(t), -P(t)) \in \mathcal{P}\_{\mathcal{X}}^{2, +}V(t, \bar{\mathfrak{x}}^{s, y; \mathcal{U}}(t)), \quad \forall t \in [s, T], \quad \mathbf{P}\text{-}a.s.$$

Then (34) holds. Let us now show (35). Fix an *<sup>ω</sup>* <sup>∈</sup> <sup>Ω</sup> such that (49) holds for any *<sup>z</sup>* <sup>∈</sup> **<sup>R</sup>***n*. For any (*p*, *<sup>P</sup>*) ∈ P2,<sup>−</sup> *<sup>x</sup> V*(*t*, *x*¯*s*,*y*;*u*¯ (*t*)), by definition (30) and (49) we have

$$\begin{split} 0 &\leq V(t, \boldsymbol{z}) - V(t, \boldsymbol{\bar{x}}(t, \boldsymbol{\omega})) - \langle p, \boldsymbol{z} - \boldsymbol{\bar{x}}^{\sf s, \boldsymbol{y}; \mathbb{R}}(t, \boldsymbol{\omega}) \rangle - \frac{1}{2} (\boldsymbol{z} - \boldsymbol{\bar{x}}^{\sf s, \boldsymbol{y}; \mathbb{R}}(t, \boldsymbol{\omega}))^{\top} P (\boldsymbol{z} - \boldsymbol{\bar{x}}^{\sf s, \boldsymbol{y}; \mathbb{R}}(t, \boldsymbol{\omega})) \\ &= - \langle p(t, \boldsymbol{\omega}) + p, \boldsymbol{z} - \boldsymbol{\bar{x}}^{\sf s, \boldsymbol{y}; \mathbb{R}}(t, \boldsymbol{\omega}) \rangle - \frac{1}{2} (\boldsymbol{z} - \boldsymbol{\bar{x}}^{\sf s, \boldsymbol{y}; \mathbb{R}}(t, \boldsymbol{\omega}))^{\top} \left( P(t, \boldsymbol{\omega}) + P \right) (\boldsymbol{z} - \boldsymbol{\bar{x}}^{\sf s, \boldsymbol{y}; \mathbb{R}}(t, \boldsymbol{\omega})) \\ &+ o \left( |\boldsymbol{z} - \boldsymbol{\bar{x}}^{\sf s, \boldsymbol{y}; \mathbb{R}}(t, \boldsymbol{\omega})|^{2} \right). \end{split}$$

Then, it is necessary that

$$p = -p(t), \quad P \le -P(t), \quad \mathbf{P}\text{-}a.s.$$

Thus, (35) holds. (36) is immediate. The proof is complete.

**Remark 3.2** It is interesting to note that if *<sup>V</sup>* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*), then (34)∼(35) reduce to

$$\begin{cases} V\_{\mathcal{X}}(t, \bar{\mathfrak{x}}^{\mathsf{s}, \mathsf{y}; \mathsf{fl}}(t)) = -p(t), \\ V\_{\mathcal{X}\mathcal{X}}(t, \bar{\mathfrak{x}}^{\mathsf{s}, \mathsf{y}; \mathsf{fl}}(t)) \le -P(t), \quad \forall t \in [\mathsf{s}, T]\_{\mathsf{f}} \quad \mathbf{P}\text{-}a.s. \end{cases} \tag{50}$$

We point out that in our jump diffusion setting the strict inequality *Vxx*(*t*, *x*¯*s*,*y*;*u*¯ (*t*)) < −*P*(*t*) may happen in some cases, as shown in the following example.

**Example 3.1** Consider the following linear stochastic control system with Poisson jumps

 $(n=1):$ 
$$\begin{cases} d\boldsymbol{x}^{s,\boldsymbol{y};\boldsymbol{\mu}}(t) = \boldsymbol{u}(t)dt + \delta\boldsymbol{u}(t)\tilde{\boldsymbol{N}}(dt), \quad t \in [s, T],\\ \boldsymbol{x}^{s,\boldsymbol{y};\boldsymbol{\mu}}(s) = \boldsymbol{y}, \end{cases}$$

for some *<sup>δ</sup>* �<sup>=</sup> 0. Here *<sup>N</sup>* is a Poisson process with the intensity *<sup>λ</sup>dt* and *<sup>N</sup>*˜ (*dt*) :<sup>=</sup> *<sup>N</sup>*(*dt*) <sup>−</sup> *λdt*(*λ* > 0) is the compensated martingale measure. The quadratic cost functional is taken as

$$J(s, y; \mu(\cdot)) = \frac{1}{2} \mathbb{E} \int\_s^T \left( \frac{1}{\lambda} - \frac{1 - \lambda}{\lambda} e^{\frac{t - T}{\delta^2}} \right) x^{s, y; \mu}(t)^2 dt.$$

#### 18 Will-be-set-by-IN-TECH 136 Stochastic Modeling and Control Stochastic Control for Jump Diffusions <sup>9</sup> <sup>19</sup>

Define

$$
\phi(t) = \delta^2 \left( 1 - e^{\frac{t-T}{\delta^2}} \right), \quad \forall t \in [0, T].
$$

G(*t*, *x*, *u*) :<sup>=</sup> <sup>H</sup>(*t*, *<sup>x</sup>*, *<sup>u</sup>*) + <sup>1</sup>

> + 1 2 tr

+ 1 2 tr **E** 

− 1 2 tr **E**  2 tr

≡ �*p*(*t*), *b*(*t*, *x*, *u*)� + tr{*q*(*t*)�*σ*(*t*, *x*, *u*)} +

the above <sup>G</sup>-function. For a <sup>Ψ</sup> <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)), we have

<sup>=</sup> *<sup>H</sup>*(*t*, *<sup>x</sup>*, *<sup>u</sup>*, *<sup>p</sup>*(*t*), *<sup>q</sup>*(*t*), *<sup>γ</sup>*(*t*, ·)) <sup>−</sup> <sup>1</sup>

defined associated with only <sup>Ψ</sup> <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)).

*Bochner integrable. We say that t is a right Lesbesgue point of z if*

lim *<sup>h</sup>*→0<sup>+</sup> 1 *h*  *t*+*h t*

<sup>G</sup>(*t*, *<sup>x</sup>*, *<sup>u</sup>*) :<sup>=</sup> *<sup>G</sup>*(*t*, *<sup>x</sup>*, *<sup>u</sup>*, <sup>Ψ</sup>(*t*, *<sup>x</sup>*), *<sup>p</sup>*(*t*), *<sup>P</sup>*(*t*)) + tr

− 1 2 tr *P*(*t*) **E** 

+ 1 2 tr **E**

+ **E** 

+ 1 2 tr *P*(*t*) 

+ 1 2 tr **E**

(see also in pp. 2013-2014 of [8]).

*of full measure in* [*a*, *b*]*.*

*R*(*t*,*e*)*c*¯(*t*,*e*)*c*¯(*t*,*e*)�

*<sup>P</sup>*(*t*)*σ*(*t*, *<sup>x</sup>*, *<sup>u</sup>*)*σ*(*t*, *<sup>x</sup>*, *<sup>u</sup>*)� <sup>−</sup> <sup>2</sup>*P*(*t*)*σ*(*t*, *<sup>x</sup>*, *<sup>u</sup>*)*σ*¯(*t*)�

*π*(*de*)

�*γ*(*t*,*e*), *c*(*t*, *x*, *u*,*e*)�*π*(*de*) − *f*(*t*, *x*, *u*)

*<sup>σ</sup>*(*t*, *<sup>x</sup>*, *<sup>u</sup>*)�[*q*(*t*) <sup>−</sup> *<sup>P</sup>*(*t*)*σ*¯(*t*)]

*σ*¯(*t*)*σ*¯(*t*)� +

 . *π*(*de*) 

*π*(*de*) 

 **E**

Δ*c*(*t*,*e*; *u*)Δ*c*(*t*,*e*; *u*)�*π*(*de*)

*π*(*de*) .

Stochastic Control for Jump Diff usions 137

 *π*(*de*)

*c*¯(*t*,*e*)*c*¯(*t*,*e*)�*π*(*de*)

(51)

 **E**

*<sup>P</sup>*(*t*)*c*(*t*, *<sup>x</sup>*, *<sup>u</sup>*,*e*)*c*(*t*, *<sup>x</sup>*, *<sup>u</sup>*,*e*)� <sup>−</sup> <sup>2</sup>*P*(*t*)*c*(*t*, *<sup>x</sup>*, *<sup>u</sup>*,*e*)*c*¯(*t*,*e*)�

*<sup>R</sup>*(*t*,*e*)*c*(*t*, *<sup>x</sup>*, *<sup>u</sup>*,*e*)*c*(*t*, *<sup>x</sup>*, *<sup>u</sup>*,*e*)� <sup>−</sup> <sup>2</sup>*R*(*t*,*e*)*c*(*t*, *<sup>x</sup>*, *<sup>u</sup>*,*e*)*c*¯(*t*,*e*)�

**Remark 3.3** Recall definitions of the Hamiltonian function *H* (17) and the generalized Hamiltonian function *G* (16), we can easily verify that they have the following relations to

*c*¯(*t*,*e*)*c*¯(*t*,*e*)� + Δ*c*(*t*,*e*; *u*)Δ*c*(*t*,*e*; *u*)�

Ψ(*t*, *x* + *c*(*t*, *x*, *u*,*e*)) − Ψ(*t*, *x*) + �*p*(*t*) + *γ*(*t*,*e*), *c*(*t*, *x*, *u*,*e*)�

 **E**

Note that, unlike the definition of generalized Hamiltonian function *G*, the G-function can be

We first recall a few results on *right Lesbesgue points* for functions with values in abstract spaces

**Definition 3.3** *Let Z be a Banach space and let z* : [*a*, *b*] → *Z be a measurable function that is*

**Lemma 3.1** *Let z* : [*a*, *b*] → *Z be as in Definition 3.3. Then the set of right Lesbesgue points of z is*


*R*(*t*,*e*)Δ*c*(*t*,*e*; *u*)Δ*c*(*t*,*e*; *u*)�*π*(*de*)

2 tr *P*(*t*) 

*R*(*t*,*e*)Δ*c*(*t*,*e*; *u*)Δ*c*(*t*,*e*; *u*)�*π*(*de*)

Δ*σ*(*t*; *u*)Δ*σ*(*t*; *u*)� +

For any *<sup>u</sup>*(·) ∈ U[*s*, *<sup>T</sup>*], applying Itô's formula to *<sup>φ</sup>*(*t*)*xs*,*y*;*u*(*t*)2, we have

$$\begin{split}d\phi(t)\mathbf{x}^{s,y;\mu}(t)^{2} &= \phi(t)\left(2\mathbf{x}^{s,y;\mu}(t)\boldsymbol{u}(t) + \lambda\boldsymbol{\delta}^{2}\boldsymbol{u}(t)^{2}\right)dt + \mathbf{x}^{s,y;\mu}(t)^{2}\left(\frac{\phi(t)}{\boldsymbol{\delta}^{2}} - 1\right)dt \\ &+ \phi(t)\left(2\boldsymbol{\delta}\mathbf{x}^{s,y;\mu}(t-)\boldsymbol{u}(t) + \boldsymbol{\delta}^{2}\boldsymbol{u}(t)^{2}\right)\tilde{N}(dt). \end{split}$$

Integrating from *s* to *T*, taking expectation on both sides, we have

$$\begin{aligned} 0 &= \mathbb{E}[\phi(T)x^{s,y;\mu}(T)^2] \\ &= \phi(s)y^2 + \mathbb{E}\int\_s^T \left[2\phi(t)x^{s,y;\mu}(t)u(t) + \lambda\delta^2\phi(t)u(t)^2 + x^{s,y;\mu}(t)^2(\frac{1}{\delta^2}\phi(t)-1)\right]dt. \end{aligned}$$

Thus

$$J(s, y; \mu(\cdot)) = \frac{1}{2}\phi(s)y^2 + \mathbb{E}\int\_s^T \lambda \delta^2 \phi(t) \left[\mu(t) + \frac{\varkappa^{s, y; \mu}(t)}{\lambda \delta^2}\right]^2 dt.$$

This implies that

$$\mathfrak{u}(t) = -\frac{\mathfrak{x}^{s,y;u}(t)}{\lambda \delta^2}, \quad \forall t \in [s, T]\_{\prime}$$

is a state feedback optimal control and the value function is

$$V(s,y) = \frac{1}{2}\delta^2 \left(1 - e^{\frac{s-T}{\delta^2}}\right) y^2, \quad \forall (s,y) \in [0,T] \times \mathbf{R}.$$

On the other hand, the second-order adjoint equation is

$$\begin{cases} dP(t) = \left(\frac{1}{\lambda} - \frac{1-\lambda}{\lambda} e^{\frac{t-T}{\beta^2}}\right) dt + \int\_{\mathbb{E}} R(t, e) \tilde{N}(d\boldsymbol{d}t), \quad t \in [\boldsymbol{s}, T],\\ P(T) = 0, \end{cases}$$

which implies

$$P(t) = \frac{1-\lambda}{\lambda} \delta^2 \left( 1 - e^{\frac{s-T}{\beta^2}} \right) + \frac{1}{\lambda} (t - T)\_\prime \quad \forall t \in [s, T].$$

Then the decreasing property of function *<sup>ρ</sup>*(*t*) <sup>≡</sup> *Vxx*(*t*, *<sup>x</sup>*¯*s*,*y*;*u*¯ (*t*)) + *P*(*t*) (noting *ρ*(*T*) = 0) results in

$$V\_{\text{XX}}(t, \bar{\mathbf{x}}^{s, \mathbf{y}; \mathbf{f}}(t)) = \delta^2 \left( 1 - e^{\frac{t - T}{\bar{\rho}^2}} \right) < -\frac{1 - \lambda}{\lambda} \delta^2 \left( 1 - e^{\frac{s - T}{\bar{\rho}^2}} \right) - \frac{1}{\lambda} (t - T) = -P(t), \quad \forall t \in [0, T).$$

We proceed to study the super-subjets of the value function in the time variable *t* along an optimal trajectory. Different from its deterministic counterpart (see [18]) and similar to but more complicated than its diffusion (without jumps) counterpart (see [18] or [19]), we observe that it is *not* the generalized Hamiltonian *G* that is to be maximized in the stochastic MP unless *V* is sufficiently smooth. Instead, it is the following G-function which contains an additional term than the H-function in the stochastic MP (Theorem 2.3). Associated with an optimal pair (*x*¯*s*,*y*;*u*¯ (·), *u*¯(·)), its corresponding adjoint processes (*p*(·), *q*(·), *γ*(·, ·)) and (*P*(·), *Q*(·), *R*(·, ·)) satisfying (20), we define a <sup>G</sup>-function <sup>G</sup> : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>→</sup> **<sup>R</sup>** as

$$\begin{split} &\mathcal{G}(t,x,u) \\ &:= \mathcal{H}(t,x,u) + \frac{1}{2} \text{tr}\Big{\{R(t,e)\tilde{c}(t,e)\tilde{c}(t,e)^{\top}\}\,\pi(de) \\ &\equiv \langle p(t),b(t,\mathbf{x},u)\rangle + \text{tr}\{q(t)^{\top}\sigma(t,\mathbf{x},u)\} + \int\_{\mathbb{E}}\langle\gamma(t,e),c(t,\mathbf{x},u,e)\rangle\pi(de) - f(t,\mathbf{x},u) \\ &\quad + \frac{1}{2} \text{tr}\Big{\{P(t)\sigma(t,\mathbf{x},u)\sigma(t,\mathbf{x},u)^{\top} - 2P(t)\sigma(t,\mathbf{x},u)\tilde{\sigma}(t)^{\top}\} \\ &\quad + \frac{1}{2} \text{tr}\Big{\{\int\_{\mathbb{E}}\left[P(t)c(t,\mathbf{x},u,e)c(t,\mathbf{x},u,e)^{\top} - 2P(t)c(t,\mathbf{x},u,e)\tilde{c}(t,e)^{\top}\right]\pi(de)\} \\ &\quad - \frac{1}{2} \text{tr}\Big{\{\int\_{\mathbb{E}}\left[R(t,e)c(t,\mathbf{x},u,e)c(t,\mathbf{x},u,e)^{\top} - 2R(t,e)c(t,\mathbf{x},u,e)\tilde{c}(t,e)^{\top}\right]\pi(de)\}. \end{split} \tag{51}$$

18 Will-be-set-by-IN-TECH

, ∀*t* ∈ [0, *T*].

*dt* + *xs*,*y*;*u*(*t*)<sup>2</sup>

*N*˜ (*dt*).

*<sup>u</sup>*(*t*) + *<sup>x</sup>s*,*y*;*u*(*t*) *λδ*<sup>2</sup>

*<sup>R</sup>*(*t*,*e*)*N*˜ (*dedt*), *<sup>t</sup>* <sup>∈</sup> [*s*, *<sup>T</sup>*],

*<sup>λ</sup>* (*<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*), <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [*s*, *<sup>T</sup>*].

� *φ*(*t*) *<sup>δ</sup>*<sup>2</sup> <sup>−</sup> <sup>1</sup>

*<sup>δ</sup>*<sup>2</sup> *<sup>φ</sup>*(*t*) <sup>−</sup> <sup>1</sup>

(*t*)) + *P*(*t*) (noting *ρ*(*T*) = 0)

*<sup>λ</sup>* (*<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*) = <sup>−</sup>*P*(*t*), <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, *<sup>T</sup>*).

�2 *dt*. ��*dt*.

� *dt*

1 − *e t*−*T <sup>δ</sup>*<sup>2</sup> �

2*xs*,*y*;*u*(*t*)*u*(*t*) + *λδ*2*u*(*t*)2�

� *T s*

<sup>2</sup>*δxs*,*y*;*u*(*t*−)*u*(*t*) + *<sup>δ</sup>*2*u*(*t*)2�

<sup>2</sup>*φ*(*t*)*xs*,*y*;*u*(*t*)*u*(*t*) + *λδ*2*φ*(*t*)*u*(*t*)<sup>2</sup> <sup>+</sup> *<sup>x</sup>s*,*y*;*u*(*t*)2� <sup>1</sup>

*λδ*2*φ*(*t*) �

*λδ*<sup>2</sup> , <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [*s*, *<sup>T</sup>*],

*<sup>y</sup>*2, <sup>∀</sup>(*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>**.

*φ*(*t*) = *δ*2�

For any *<sup>u</sup>*(·) ∈ U[*s*, *<sup>T</sup>*], applying Itô's formula to *<sup>φ</sup>*(*t*)*xs*,*y*;*u*(*t*)2, we have

�

Integrating from *s* to *T*, taking expectation on both sides, we have

2

is a state feedback optimal control and the value function is

2 *δ*2� 1 − *e s*−*T <sup>δ</sup>*<sup>2</sup> �

*<sup>λ</sup>* <sup>−</sup> <sup>1</sup> <sup>−</sup> *<sup>λ</sup> <sup>λ</sup> <sup>e</sup> t*−*T δ*2 � *dt* + � **E**

*<sup>λ</sup> <sup>δ</sup>*2�

Then the decreasing property of function *<sup>ρ</sup>*(*t*) <sup>≡</sup> *Vxx*(*t*, *<sup>x</sup>*¯*s*,*y*;*u*¯

satisfying (20), we define a <sup>G</sup>-function <sup>G</sup> : [0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***<sup>n</sup>* <sup>×</sup> **<sup>U</sup>** <sup>→</sup> **<sup>R</sup>** as

1 − *e s*−*T <sup>δ</sup>*<sup>2</sup> � + 1

<sup>&</sup>lt; <sup>−</sup><sup>1</sup> <sup>−</sup> *<sup>λ</sup>*

*<sup>λ</sup> <sup>δ</sup>*2�

1 − *e s*−*T <sup>δ</sup>*<sup>2</sup> � − 1

(·), *u*¯(·)), its corresponding adjoint processes (*p*(·), *q*(·), *γ*(·, ·)) and (*P*(·), *Q*(·), *R*(·, ·))

We proceed to study the super-subjets of the value function in the time variable *t* along an optimal trajectory. Different from its deterministic counterpart (see [18]) and similar to but more complicated than its diffusion (without jumps) counterpart (see [18] or [19]), we observe that it is *not* the generalized Hamiltonian *G* that is to be maximized in the stochastic MP unless *V* is sufficiently smooth. Instead, it is the following G-function which contains an additional term than the H-function in the stochastic MP (Theorem 2.3). Associated with an optimal pair

*<sup>V</sup>*(*s*, *<sup>y</sup>*) = <sup>1</sup>

On the other hand, the second-order adjoint equation is

� 1

*<sup>P</sup>*(*t*) = <sup>1</sup> <sup>−</sup> *<sup>λ</sup>*

1 − *e t*−*T <sup>δ</sup>*<sup>2</sup> �

*φ*(*s*)*y*<sup>2</sup> + **E**

*<sup>u</sup>*¯(*t*) = <sup>−</sup> *<sup>x</sup>s*,*y*;*u*(*t*)

+ *φ*(*t*) �

*dφ*(*t*)*xs*,*y*;*u*(*t*)<sup>2</sup> = *φ*(*t*)

0 = **E**[*φ*(*T*)*xs*,*y*;*u*(*T*)2]

� *T s* �

*<sup>J</sup>*(*s*, *<sup>y</sup>*; *<sup>u</sup>*(·)) = <sup>1</sup>

= *φ*(*s*)*y*<sup>2</sup> + **E**

⎧ ⎨ ⎩

*dP*(*t*) =

*P*(*T*) = 0,

Define

Thus

This implies that

which implies

results in

(*x*¯*s*,*y*;*u*¯

*Vxx*(*t*, *x*¯

*s*,*y*;*u*¯

(*t*)) = *δ*2�

**Remark 3.3** Recall definitions of the Hamiltonian function *H* (17) and the generalized Hamiltonian function *G* (16), we can easily verify that they have the following relations to the above <sup>G</sup>-function. For a <sup>Ψ</sup> <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)), we have

<sup>G</sup>(*t*, *<sup>x</sup>*, *<sup>u</sup>*) :<sup>=</sup> *<sup>G</sup>*(*t*, *<sup>x</sup>*, *<sup>u</sup>*, <sup>Ψ</sup>(*t*, *<sup>x</sup>*), *<sup>p</sup>*(*t*), *<sup>P</sup>*(*t*)) + tr *<sup>σ</sup>*(*t*, *<sup>x</sup>*, *<sup>u</sup>*)�[*q*(*t*) <sup>−</sup> *<sup>P</sup>*(*t*)*σ*¯(*t*)] − 1 2 tr *P*(*t*) **E** *c*¯(*t*,*e*)*c*¯(*t*,*e*)� + Δ*c*(*t*,*e*; *u*)Δ*c*(*t*,*e*; *u*)� *π*(*de*) + 1 2 tr **E** *R*(*t*,*e*)Δ*c*(*t*,*e*; *u*)Δ*c*(*t*,*e*; *u*)�*π*(*de*) + **E** Ψ(*t*, *x* + *c*(*t*, *x*, *u*,*e*)) − Ψ(*t*, *x*) + �*p*(*t*) + *γ*(*t*,*e*), *c*(*t*, *x*, *u*,*e*)� *π*(*de*) <sup>=</sup> *<sup>H</sup>*(*t*, *<sup>x</sup>*, *<sup>u</sup>*, *<sup>p</sup>*(*t*), *<sup>q</sup>*(*t*), *<sup>γ</sup>*(*t*, ·)) <sup>−</sup> <sup>1</sup> 2 tr *P*(*t*) *σ*¯(*t*)*σ*¯(*t*)� + **E** *c*¯(*t*,*e*)*c*¯(*t*,*e*)�*π*(*de*) + 1 2 tr *P*(*t*) Δ*σ*(*t*; *u*)Δ*σ*(*t*; *u*)� + **E** Δ*c*(*t*,*e*; *u*)Δ*c*(*t*,*e*; *u*)�*π*(*de*) + 1 2 tr **E** *R*(*t*,*e*)Δ*c*(*t*,*e*; *u*)Δ*c*(*t*,*e*; *u*)�*π*(*de*) .

Note that, unlike the definition of generalized Hamiltonian function *G*, the G-function can be defined associated with only <sup>Ψ</sup> <sup>∈</sup> *<sup>C</sup>*([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)).

We first recall a few results on *right Lesbesgue points* for functions with values in abstract spaces (see also in pp. 2013-2014 of [8]).

**Definition 3.3** *Let Z be a Banach space and let z* : [*a*, *b*] → *Z be a measurable function that is Bochner integrable. We say that t is a right Lesbesgue point of z if*

$$\lim\_{h \to 0^+} \frac{1}{h} \int\_t^{t+h} |z(r) - z(t)|\_Z dr = 0.$$

**Lemma 3.1** *Let z* : [*a*, *b*] → *Z be as in Definition 3.3. Then the set of right Lesbesgue points of z is of full measure in* [*a*, *b*]*.*

#### 20 Will-be-set-by-IN-TECH 138 Stochastic Modeling and Control Stochastic Control for Jump Diffusions <sup>10</sup> <sup>21</sup>

The second main result in this subsection is the following.

**Theorem 3.3** *Suppose that* **(H1)**∼**(H3)** *hold and let* (*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> be fixed. Let* (*x*¯*s*,*y*;*u*¯ (·), *u*¯(·)) *be an optimal pair of our stochastic control problem. Let* (*p*(·), *q*(·), *γ*(·, ·)) *and* (*P*(·), *Q*(·), *R*(·, ·)) *are first-order and second-order adjoint processes, respectively. Then we have*

$$\mathcal{G}(t, \bar{\mathfrak{x}}^{\mathbf{s}, \mathbf{y}; \mathfrak{A}}(t), \mathfrak{u}(t)) \in \mathcal{P}\_{t+}^{1,+} V(t, \bar{\mathfrak{x}}^{\mathbf{s}, \mathbf{y}; \mathfrak{A}}(t)), \quad a.e. t \in [\mathbf{s}, T], \mathbf{P}\text{-a.s.},\tag{52}$$

where ⎧

*ετ*1(*r*) :=

*ετ*2(*r*) :=

*ετ*3(*r*, ·) :=

*ετ*4(*r*) :=

*ετ*5(*r*) :=

*ετ*6(*r*, ·) :=

� 1 0 � *bx*(*r*, *x*¯

� 1 0 � *σx*(*r*, *x*¯

� 1 0 � *cx*(*r*, *x*¯

� 1 0

� 1 0

� 1 0

function *<sup>δ</sup>* : [0, <sup>∞</sup>) <sup>→</sup> [0, <sup>∞</sup>), with *<sup>δ</sup>*(*r*)

⎧

**E** � � *<sup>T</sup> τ*

**E** � � *<sup>T</sup> τ*

**E** � � *<sup>T</sup> τ*

**E** � � *<sup>T</sup> τ*

**E** � � *<sup>T</sup> τ*

**E** � � *<sup>T</sup> τ*

(*t*)) ≤ **E**

(*t*)) ≤ **E**

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

*V*(*τ*, *x*¯

*V*(*τ*, *x*¯

⎧ ⎪⎨

⎪⎩

Taking **<sup>E</sup>**(·|F*<sup>s</sup>*

*s*,*y*;*u*¯

*s*,*y*;*u*¯

*V*(*t*, *x*¯ *s*,*y*;*u*¯ *s*,*y*;*u*¯

*s*,*y*;*u*¯

*s*,*y*;*u*¯

(<sup>1</sup> <sup>−</sup> *<sup>θ</sup>*)*ξτ*(*r*)��

(<sup>1</sup> <sup>−</sup> *<sup>θ</sup>*)*ξτ*(*r*)��

(<sup>1</sup> <sup>−</sup> *<sup>θ</sup>*)*ξτ*(*r*−)��





� � *<sup>T</sup> τ*

� � *<sup>T</sup> τ*

(*t*, *ω*0)) = **E**

*<sup>t</sup>* ) on both sides and noting that <sup>F</sup>*<sup>s</sup>*



(*r*) + *θξτ*(*r*), *<sup>u</sup>*¯(*r*)) <sup>−</sup> ¯

*bxx*(*r*, *x*¯

*σxx*(*r*, *x*¯

<sup>2</sup>*dr*� �F*s t* �

<sup>2</sup>*dr*� �F*s t* �

<sup>2</sup>*dr*� �F*s t* �

<sup>2</sup>*dr*� �F*s t* �

<sup>L</sup><sup>2</sup> *dr*� �F*s t* �

<sup>L</sup><sup>2</sup> *dr*� �F*s t* �

Choose a common subset Ω<sup>0</sup> ⊆ Ω with *P*(Ω0) = 1 such that for any *ω*<sup>0</sup> ∈ Ω0,

� � *<sup>T</sup> s* ¯

Note that *u*¯(·)|[*τ*,*T*] ∈ U[*τ*, *T*], **P**-*a*.*s*. Thus by the definition of the value function *V*, we have

*<sup>f</sup>*(*r*, *<sup>x</sup>τ*(*r*), *<sup>u</sup>*¯(*r*))*dr* + *<sup>h</sup>*(*xτ*(*T*))�

*<sup>f</sup>*(*r*, *<sup>x</sup>τ*(*r*), *<sup>u</sup>*¯(*r*))*dr* + *<sup>h</sup>*(*xτ*(*T*))�

*f*(*r*)*dr* + *h*(*x*¯

(54),(57) are satisfied for any rational *τ* > *t*, and *u*¯(·)|[*τ*,*T*] ∈ U[*τ*, *T*].

*cxx*(*r*, *x*¯

(*r*) + *θξτ*(*r*), *u*¯(*r*)) − *σ*¯*x*(*r*)

*s*,*y*;*u*¯

*s*,*y*;*u*¯

*s*,*y*;*u*¯

Similar to the proof of (41) and (42), there exists a deterministic continuous and increasing

*<sup>r</sup>* → 0 as *r* → 0, such that

(*ω*) ≤ *δ*(|*τ* − *t*|

(*ω*) ≤ *δ*(|*τ* − *t*|

*<sup>t</sup>* ⊆ F*<sup>s</sup>*

*s*,*y*;*u*¯ (*T*))� �F*s t* � (*ω*0),

(*ω*) ≤ *δ*(|*τ* − *t*|

(*r*−) + *θξτ*(*r*−), *u*¯(*r*), ·) − *c*¯*x*(*r*, ·)

*bx*(*r*) � *ξτ*(*r*)*dθ*,

> � *ξτ*(*r*)*dθ*,

(*r*) + *θξτ*(*r*), *<sup>u</sup>*¯(*r*)) <sup>−</sup> ¯

(*r*) + *θξτ*(*r*), *u*¯(*r*)) − *σ*¯*xx*(*r*)

(*ω*) ≤ *δ*(|*τ* − *t*|), **P**-*a*.*s*.*ω*,

(*ω*) ≤ *δ*(|*τ* − *t*|), **P**-*a*.*s*.*ω*,

(*ω*) ≤ *δ*(|*τ* − *t*|), **P**-*a*.*s*.*ω*,

<sup>2</sup>), **P**-*a*.*s*.*ω*,

<sup>2</sup>), **P**-*a*.*s*.*ω*,

�F*s τ* �

*<sup>τ</sup>*, we conclude that

�F*s t* �

<sup>2</sup>), **P**-*a*.*s*.*ω*.

, ∀*τ* ∈ (*t*, *T*], **P**-*a*.*s*.

, ∀*τ* ∈ (*t*, *T*], **P**-*a*.*s*. (58)

�

(*r*−) + *θξτ*(*r*−), *u*¯(*r*), ·) − *c*¯*xx*(*r*, ·)

*ξτ*(*r*−)*dθ*,

*bxx*(*r*) � *ξτ*(*r*)*dθ*,

> � *ξτ*(*r*)*dθ*,

> > �

Stochastic Control for Jump Diff usions 139

*ξτ*(*r*−)*dθ*.

(57)

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

*where the* G*-function is defined by (51).*

*Proof* For any *t* ∈ (*s*, *T*), take *τ* ∈ (*t*, *T*]. Denote by *xτ*(·) the solution of the following SDEP

$$\begin{split} \mathbf{x}\_{\tau}(\boldsymbol{r}) &= \mathbf{\bar{x}}^{\boldsymbol{\theta},\boldsymbol{y},\boldsymbol{\theta}}(\boldsymbol{t}) + \int\_{\tau}^{\boldsymbol{r}} b(\boldsymbol{\theta}, \boldsymbol{x}\_{\tau}(\boldsymbol{\theta}), \boldsymbol{\bar{u}}(\boldsymbol{\theta})) d\boldsymbol{\theta} + \int\_{\tau}^{\boldsymbol{r}} \boldsymbol{\sigma}(\boldsymbol{\theta}, \boldsymbol{x}\_{\tau}(\boldsymbol{\theta}), \boldsymbol{\bar{u}}(\boldsymbol{\theta})) d\boldsymbol{\mathcal{W}}(\boldsymbol{\theta}) \\ &+ \int\_{\mathcal{E}} \int\_{\tau}^{\boldsymbol{r}} \boldsymbol{c}(\boldsymbol{\theta}, \boldsymbol{x}\_{\tau}(\boldsymbol{\theta} - \boldsymbol{\\_}), \boldsymbol{\bar{u}}(\boldsymbol{\theta}), \boldsymbol{e}) \tilde{\mathbf{N}}(\boldsymbol{d} \boldsymbol{c} d\boldsymbol{\theta}), \quad \boldsymbol{r} \in [\boldsymbol{\tau}, T]. \end{split} \tag{53}$$

Set *ξτ*(*r*) :<sup>=</sup> *<sup>x</sup>τ*(*r*) <sup>−</sup> *<sup>x</sup>*¯*s*,*y*;*u*¯ (*r*), *τ* ≤ *r* ≤ *T*. Working under the new probability measure **<sup>P</sup>**(·|F*<sup>s</sup> <sup>τ</sup>*)(*ω*), we have the following estimate by (10):

$$\mathbb{E}\left[\sup\_{\tau\le r\le T}|\xi\_{\tau}(r)|^{2}|\mathcal{F}\_{\tau}^{\sf s}\right](\omega) \le \mathbb{C}|\widetilde{\pi}^{\sf s,\sf y;\widetilde{\boldsymbol{\mu}}}(\tau,\omega) - \widetilde{\pi}^{\sf s,\sf y;\widetilde{\boldsymbol{\mu}}}(t,\omega)|^{2}, \quad \textsf{P-a.s.}\omega.$$

Taking **<sup>E</sup>**(·|F*<sup>s</sup> <sup>t</sup>* )(*ω*) on both sides and noting that <sup>F</sup>*<sup>s</sup> <sup>t</sup>* ⊆ F*<sup>s</sup> <sup>τ</sup>*, by (11), we obtain

$$\mathbb{E}\left[\sup\_{\tau\le r\le T} |\xi\_{\tau}(r)|^{2} |\mathcal{F}\_{t}^{s}\right](\omega) \le \mathbb{C}|\tau - t|\_{\prime} \quad \text{P-a.s.}\omega.\tag{54}$$

The process *ξτ*(·) satisfies the following variational equations:

$$\begin{cases} d\tilde{\xi}\_{\mathsf{T}}(r) = \bar{b}\_{\mathsf{X}}(r)\tilde{\xi}\_{\mathsf{T}}(r)dr + \bar{\sigma}\_{\mathsf{X}}(r)\tilde{\xi}\_{\mathsf{T}}(r)dW(r) + \int\_{\mathsf{E}} \tilde{\varepsilon}\_{\mathsf{X}}(r,e)\tilde{\xi}\_{\mathsf{T}}(r-)\tilde{N}(dedr) \\\\ \qquad + \varepsilon\_{\mathsf{T}1}(r)dr + \varepsilon\_{\mathsf{T}2}(r)dW(r) + \int\_{\mathsf{E}} \varepsilon\_{\mathsf{T}3}(r,e)\tilde{N}(dedr), \quad r \in [t, T], \\\\ \tilde{\xi}\_{\mathsf{T}}(\mathsf{r}) = -\int\_{t}^{\mathsf{T}} \bar{b}(r)dr - \int\_{t}^{\mathsf{T}} \bar{\sigma}(r)dW(r) - \int\_{\mathsf{E}} \int\_{t}^{\mathsf{T}} \bar{\varepsilon}(r,e)\tilde{N}(dedr), \end{cases} (55)$$

and

$$\begin{cases} d\xi\_{\tau}(r) = \left\{\bar{b}\_{\mathbf{x}}(r)\xi\_{\tau}(r) + \frac{1}{2}\xi\_{\tau}(r)^{\top}\bar{b}\_{\mathbf{x}\mathbf{x}}(r)\xi\_{\tau}(r)\right\} dr \\\\ \qquad \quad + \left\{\bar{\sigma}\_{\mathbf{x}}(r)\xi\_{\tau}(r) + \frac{1}{2}\xi\_{\tau}(r)^{\top}\bar{\sigma}\_{\mathbf{x}\mathbf{x}}(r)\xi\_{\tau}(r)\right\} dW(r) \\\\ \qquad \quad \quad + \int\_{\mathbf{E}}\left\{\bar{\varepsilon}\_{\mathbf{x}}(r,e)\xi\_{\tau}(r-) + \frac{1}{2}\xi\_{\tau}(r-)^{\top}\bar{\varepsilon}\_{\mathbf{x}\mathbf{x}}(r,e)\xi\_{\tau}(r-)\right\} \bar{N}(dedr) \\\\ \qquad \quad \quad \quad \quad + \varepsilon\_{\tau4}(r)dr + \varepsilon\_{\tau5}(r)dW(r) + \int\_{\mathbf{E}}\varepsilon\_{\tau6}(r,e)\bar{N}(dedr), \quad r \in [t,T], \\\\ \xi\_{\tau}(\tau) = -\int\_{t}^{\tau}\bar{b}(r)dr - \int\_{t}^{\tau}\bar{\sigma}(r)dW(r) - \int\_{\mathbf{E}}\int\_{t}^{\tau}\bar{\varepsilon}(r,e)\bar{N}(dedr), \end{cases} (56)$$

where

20 Will-be-set-by-IN-TECH

**Theorem 3.3** *Suppose that* **(H1)**∼**(H3)** *hold and let* (*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> be fixed. Let*

(*P*(·), *Q*(·), *R*(·, ·)) *are first-order and second-order adjoint processes, respectively. Then we have*

*Proof* For any *t* ∈ (*s*, *T*), take *τ* ∈ (*t*, *T*]. Denote by *xτ*(·) the solution of the following SDEP

*b*(*θ*, *xτ*(*θ*), *u*¯(*θ*))*dθ* +

*<sup>t</sup>*<sup>+</sup> *V*(*t*, *x*¯

(·), *u*¯(·)) *be an optimal pair of our stochastic control problem. Let* (*p*(·), *q*(·), *γ*(·, ·)) *and*

*s*,*y*;*u*¯

*<sup>c</sup>*(*θ*, *<sup>x</sup>τ*(*θ*−), *<sup>u</sup>*¯(*θ*),*e*)*N*˜ (*dedθ*), *<sup>r</sup>* <sup>∈</sup> [*τ*, *<sup>T</sup>*].

*s*,*y*;*u*¯

(*τ*, *ω*) − *x*¯

*<sup>t</sup>* ⊆ F*<sup>s</sup>*

� **E**

� **E**

> � **E** � *τ t*

> > � *dr*

> > > � *dW*(*r*)

*ξτ*(*r*−)�*c*¯*xx*(*r*,*e*)*ξτ*(*r*−)

*σ*¯(*r*)*dW*(*r*) −

*bxx*(*r*)*ξτ*(*r*)

*ξτ*(*r*)�*σ*¯*xx*(*r*)*ξτ*(*r*)

� **E**

> � **E** � *τ t*

2

*σ*¯(*r*)*dW*(*r*) −

� *r τ*

(*r*), *τ* ≤ *r* ≤ *T*. Working under the new probability measure

*s*,*y*;*u*¯

(*t*, *ω*)|

*<sup>τ</sup>*, by (11), we obtain

(*ω*) ≤ *C*|*τ* − *t*|, **P**-*a*.*s*.*ω*. (54)

*<sup>c</sup>*¯*x*(*r*,*e*)*ξτ*(*r*−)*N*˜ (*dedr*)

*ετ*3(*r*,*e*)*N*˜ (*dedr*), *<sup>r</sup>* <sup>∈</sup> [*t*, *<sup>T</sup>*],

�

*ετ*6(*r*,*e*)*N*˜ (*dedr*), *<sup>r</sup>* <sup>∈</sup> [*t*, *<sup>T</sup>*],

*c*¯(*r*,*e*)*N*˜ (*dedr*),

*N*˜ (*dedr*)

*c*¯(*r*,*e*)*N*˜ (*dedr*),

(*t*)), *a*.*e*.*t* ∈ [*s*, *T*], **P***-a*.*s*., (52)

*σ*(*θ*, *xτ*(*θ*), *u*¯(*θ*))*dW*(*θ*)

2, **P**-*a*.*s*.*ω*.

(53)

(55)

(56)

The second main result in this subsection is the following.

(*t*), *<sup>u</sup>*¯(*t*)) ∈ P1,<sup>+</sup>

G(*t*, *x*¯ *s*,*y*;*u*¯

*where the* G*-function is defined by (51).*

*xτ*(*r*) = *x*¯

Set *ξτ*(*r*) :<sup>=</sup> *<sup>x</sup>τ*(*r*) <sup>−</sup> *<sup>x</sup>*¯*s*,*y*;*u*¯

**E** � sup *τ*≤*r*≤*T*

*s*,*y*;*u*¯ (*t*) + � *r τ*

*<sup>τ</sup>*)(*ω*), we have the following estimate by (10):

<sup>2</sup>|F*<sup>s</sup> τ* �

The process *ξτ*(·) satisfies the following variational equations:

*bx*(*r*)*ξτ*(*r*) + <sup>1</sup>

*<sup>t</sup>* )(*ω*) on both sides and noting that <sup>F</sup>*<sup>s</sup>*


(*ω*) ≤ *C*|*x*¯

<sup>2</sup>|F*<sup>s</sup> t* �

*bx*(*r*)*ξτ*(*r*)*dr* + *σ*¯*x*(*r*)*ξτ*(*r*)*dW*(*r*) +

+ *ετ*1(*r*)*dr* + *ετ*2(*r*)*dW*(*r*) +

� *τ t*

2

*<sup>σ</sup>*¯*x*(*r*)*ξτ*(*r*) + <sup>1</sup>

*ξτ*(*r*)� ¯

2

*<sup>c</sup>*¯*x*(*r*,*e*)*ξτ*(*r*−) + <sup>1</sup>

+ *ετ*4(*r*)*dr* + *ετ*5(*r*)*dW*(*r*) +

� *τ t*


**E** � sup *τ*≤*r*≤*T*

> � *τ t* ¯ *b*(*r*)*dr* −

� ¯

> + �

+ � **E** �

� *τ t* ¯ *b*(*r*)*dr* −

+ � **E** � *r τ*

(*x*¯*s*,*y*;*u*¯

**<sup>P</sup>**(·|F*<sup>s</sup>*

Taking **<sup>E</sup>**(·|F*<sup>s</sup>*

⎧

*dξτ*(*r*) = ¯

*ξτ*(*τ*) = −

*dξτ*(*r*) =

*ξτ*(*τ*) = −

⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

and

⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *ετ*1(*r*) := � 1 0 � *bx*(*r*, *x*¯ *s*,*y*;*u*¯ (*r*) + *θξτ*(*r*), *<sup>u</sup>*¯(*r*)) <sup>−</sup> ¯ *bx*(*r*) � *ξτ*(*r*)*dθ*, *ετ*2(*r*) := � 1 0 � *σx*(*r*, *x*¯ *s*,*y*;*u*¯ (*r*) + *θξτ*(*r*), *u*¯(*r*)) − *σ*¯*x*(*r*) � *ξτ*(*r*)*dθ*, *ετ*3(*r*, ·) := � 1 0 � *cx*(*r*, *x*¯ *s*,*y*;*u*¯ (*r*−) + *θξτ*(*r*−), *u*¯(*r*), ·) − *c*¯*x*(*r*, ·) � *ξτ*(*r*−)*dθ*, *ετ*4(*r*) := � 1 0 (<sup>1</sup> <sup>−</sup> *<sup>θ</sup>*)*ξτ*(*r*)�� *bxx*(*r*, *x*¯ *s*,*y*;*u*¯ (*r*) + *θξτ*(*r*), *<sup>u</sup>*¯(*r*)) <sup>−</sup> ¯ *bxx*(*r*) � *ξτ*(*r*)*dθ*, *ετ*5(*r*) := � 1 0 (<sup>1</sup> <sup>−</sup> *<sup>θ</sup>*)*ξτ*(*r*)�� *σxx*(*r*, *x*¯ *s*,*y*;*u*¯ (*r*) + *θξτ*(*r*), *u*¯(*r*)) − *σ*¯*xx*(*r*) � *ξτ*(*r*)*dθ*, *ετ*6(*r*, ·) := � 1 0 (<sup>1</sup> <sup>−</sup> *<sup>θ</sup>*)*ξτ*(*r*−)�� *cxx*(*r*, *x*¯ *s*,*y*;*u*¯ (*r*−) + *θξτ*(*r*−), *u*¯(*r*), ·) − *c*¯*xx*(*r*, ·) � *ξτ*(*r*−)*dθ*.

Similar to the proof of (41) and (42), there exists a deterministic continuous and increasing function *<sup>δ</sup>* : [0, <sup>∞</sup>) <sup>→</sup> [0, <sup>∞</sup>), with *<sup>δ</sup>*(*r*) *<sup>r</sup>* → 0 as *r* → 0, such that

$$\begin{cases} \mathbb{E}\left[\int\_{\tau}^{T} |\varepsilon\_{1}(r)|^{2} dr |\mathcal{F}\_{l}^{s}| \right](\omega) \leq \delta(|\tau - t|), \quad \mathbf{P}\text{-a.s.}\omega, \\ \mathbb{E}\left[\int\_{\tau}^{T} |\varepsilon\_{12}(r)|^{2} dr |\mathcal{F}\_{l}^{s}| \right](\omega) \leq \delta(|\tau - t|), \quad \mathbf{P}\text{-a.s.}\omega, \\ \mathbb{E}\left[\int\_{\tau}^{T} ||\varepsilon\_{73}(r\_{\prime} \cdot |)||\_{\mathcal{L}^{2}}^{2} dr |\mathcal{F}\_{l}^{s}| \right](\omega) \leq \delta(|\tau - t|), \quad \mathbf{P}\text{-a.s.}\omega, \\ \mathbb{E}\left[\int\_{\tau}^{T} |\varepsilon\_{74}(r)|^{2} dr |\mathcal{F}\_{l}^{s}| \right](\omega) \leq \delta(|\tau - t|^{2}), \quad \mathbf{P}\text{-a.s.}\omega, \\ \mathbb{E}\left[\int\_{\tau}^{T} |\varepsilon\_{75}(r)|^{2} dr |\mathcal{F}\_{l}^{s}| \right](\omega) \leq \delta(|\tau - t|^{2}), \quad \mathbf{P}\text{-a.s.}\omega, \\ \mathbb{E}\left[\int\_{\tau}^{T} ||\varepsilon\_{76}(r\_{\prime} \cdot |)||\_{\mathcal{L}^{2}}^{2} dr |\mathcal{F}\_{l}^{s}| \right](\omega) \leq \delta(|\tau - t|^{2}), \quad \mathbf{P}\text{-a.s.}\omega. \end{cases} \tag{57}$$

Note that *u*¯(·)|[*τ*,*T*] ∈ U[*τ*, *T*], **P**-*a*.*s*. Thus by the definition of the value function *V*, we have

$$V(\boldsymbol{\tau}, \boldsymbol{\mathfrak{x}}^{\mathbf{s}, \mathbf{y}; \boldsymbol{\tilde{u}}}(t)) \leq \mathbb{E} \Big[ \int\_{\tau}^{T} f(\boldsymbol{r}, \boldsymbol{\mathfrak{x}}\_{\tau}(\boldsymbol{r}), \boldsymbol{\tilde{u}}(\boldsymbol{r})) d\boldsymbol{r} + h(\boldsymbol{\mathfrak{x}}\_{\tau}(\boldsymbol{T})) \Big| \mathcal{F}\_{\tau}^{\mathbf{s}} \Big]\_{\prime} \quad \forall \tau \in (t, T], \mathbf{P}\text{-}a.s.$$

Taking **<sup>E</sup>**(·|F*<sup>s</sup> <sup>t</sup>* ) on both sides and noting that <sup>F</sup>*<sup>s</sup> <sup>t</sup>* ⊆ F*<sup>s</sup> <sup>τ</sup>*, we conclude that

$$V(\mathbf{r}, \mathbf{\tilde{x}}^{\mathbf{s}; \mathbf{y}; \mathbf{\tilde{z}}}(t)) \le \mathbb{E} \left[ \int\_{\tau}^{T} f(r, \mathbf{x}\_{\tau}(r), \mathbf{\tilde{u}}(r)) dr + h(\mathbf{x}\_{\tau}(T)) \Big| \mathcal{F}\_{t}^{\mathbf{s}} \right], \quad \forall \tau \in (t, T], \mathbf{P}\text{-a.s.} \tag{58}$$

Choose a common subset Ω<sup>0</sup> ⊆ Ω with *P*(Ω0) = 1 such that for any *ω*<sup>0</sup> ∈ Ω0,

$$\begin{cases} V(t, \mathfrak{x}^{\mathfrak{s}, \mathfrak{z}; \mathfrak{z}}(t, \omega\_0)) = \mathbb{E} \left[ \int\_s^T \bar{f}(r) dr + h(\mathfrak{x}^{\mathfrak{s}, \mathfrak{z}; \mathfrak{z}}(T)) \Big| \mathcal{F}\_t^{\mathfrak{s}} \right](\omega\_0), \\ (54)\_\prime (57) \text{ are satisfied for any rational } \mathfrak{r} > t, \text{ and } \bar{u}(\cdot)|\_{[\mathfrak{r}, T]} \in \mathcal{U}[\mathfrak{r}, T]. \end{cases}$$

Let *ω*<sup>0</sup> ∈ Ω<sup>0</sup> be fixed, then for any rational *τ* > *t*, we have (noting (58))

*V*(*τ*, *x*¯ *s*,*y*;*u*¯ (*t*, *ω*0)) − *V*(*t*, *x*¯ *s*,*y*;*u*¯ (*t*, *ω*0)) ≤ **E** − *τ t* ¯ *f*(*r*)*dr* + *T τ* [ *<sup>f</sup>*(*r*, *<sup>x</sup>τ*(*r*), *<sup>u</sup>*¯(*r*)) <sup>−</sup> ¯ *f*(*r*)]*dr* + *h*(*xτ*(*T*)) − *h*(*x*¯ *s*,*y*;*u*¯ (*T*)) F*s t* (*ω*0) = **E** − *τ t* ¯ *f*(*r*)*dr* + *T τ* � ¯ *fx*(*r*), *ξτ*(*r*)�*dr* + �*hx*(*x*¯ *s*,*y*;*u*¯ (*T*)), *ξτ*(*T*)� + 1 2 *T τ ξτ*(*r*)� ¯ *fxx*(*r*)*ξτ*(*r*)*dr* + 1 2 *ξτ*(*T*)�*hxx*(*x*¯ *s*,*y*;*u*¯ (*T*))*ξτ*(*T*) F*s t* (*ω*0) + *o*(|*τ* − *t*|).

All the last equalities in the above three inequalities is due to the fact that the sets of right Lebesgue points have full Lebesgue measures for integrable functions by Lemma 3.1 and *t* �→

*<sup>t</sup>* is right continuous in *t*. Thus applying Itô's formula to �*p*(·), *ξτ*(·)�, by (55), (18) we have

 **E** *τ t*

*c*¯(*r*,*e*)*N*˜ (*dedr*)

*c*¯*x*(*r*,*e*)*γ*(*r*,*e*)*π*(*de*)

 *τ t* ¯ *b*(*r*)*dr*

 F*s t* (*ω*0) Stochastic Control for Jump Diff usions 141

(60)

(61)

(62)

 *dr*

(*ω*0) + *o*(|*τ* − *t*|), as *τ* ↓ *t*, *a*.*e*.*t* ∈ [*s*, *T*).

F*s t* (*ω*0)

tr{*c*¯(*r*,*e*)�*P*(*t*)*c*¯(*r*,*e*)}*π*(*de*)*dr*

tr{*q*(*r*)�*σ*¯(*r*)}*dr*

�*γ*(*r*,*e*)�*c*¯(*r*,*e*)�*π*(*de*)*dr*

*R*(*t*,*e*)*c*¯(*t*,*e*)*c*¯(*t*,*e*)�

(*ω*0) + *o*(|*τ* − *t*|)

*π*(*de*) 

*π*(*de*) – containing the integrand

 F*s t* (*ω*0)

**E**

*<sup>γ</sup>*(*r*,*e*)*N*˜ (*dedr*), <sup>−</sup>

*c*¯(*r*,*e*)*N*˜ (*dedr*)

tr{*q*(*r*)�*σ*¯(*r*)}*dr*

F*s t* 

Similarly, applying Itô's formula to *ξτ*(·)�*P*(·)*ξτ*(·)�, by (56) and (19), we have

 **E** *τ t*

*s*,*y*;*u*¯

tr{*c*¯(*r*,*e*)�*P*(*t*)*c*¯(*r*,*e*)}*π*(*de*)*dr*

(*t*, *<sup>ω</sup>*0), *<sup>u</sup>*¯(*t*)) + <sup>1</sup>

 *τ t* ¯ *b*(*r*)*dr*� +

(*t*, *ω*0))

 **E** *τ t*

> 2 tr

By the same argument as in the paragraph following (49), we conclude that (62) holds for any

**Remark 3.4** As aforementioned, it is worth to point out that by comparing the result of Theorem 3.3 with those analogue in the diffusion case (see Theorem 4.7 on pp. 263 of [18]), we

*R*(*t*,*e*)*c*¯(*t*,*e*)*c*¯(*t*,*e*)�

 *τ t*

> F*s t*

(*t*, *ω*0), *u*¯(*t*)) + *o*(|*τ* − *t*|), as *τ* ↓ *t*, *a*.*e*.*t* ∈ [*s*, *T*).

*σ*¯(*r*)*dW*(*r*) −

F*s*

**E** 

= **E** 

= **E**

= **E** 

> − **E** *τ t*

**E** 

> *<sup>τ</sup> t*

= **E**

+ − *τ t* ¯

�*p*(*τ*), *ξτ*(*τ*)�

*<sup>p</sup>*(*t*), <sup>−</sup>

+ *τ t*

− *τ t*

− �*p*(*t*),

 F*s t* (*ω*0)

*<sup>q</sup>*(*r*)*dW*(*r*) +

*σ*¯(*r*)*dW*(*r*) −

 *τ t* ¯

*ξτ*(*τ*)�*P*(*τ*)*ξτ*(*τ*)

*V*(*τ*, *x*¯

− 1 2 *τ t*

− 1 2 **E** *τ t*

= (*τ* − *t*)G(*t*, *x*¯

observe that an additional term – <sup>1</sup>

 H(*t*, *x*¯

≤ (*τ* − *t*)

≤ **E** − *τ t* ¯

*s*,*y*;*u*¯

 *τ t* ¯ *b*(*r*)*dr* −

�*p*(*t*), *ξτ*(*τ*)� + �*p*(*τ*) − *p*(*t*), *ξτ*(*τ*)�

 *τ t*

*bx*(*r*)*p*(*r*) + *<sup>σ</sup>*¯*x*(*r*)*q*(*r*) +

**E** *τ t*

*t*

 **E** *τ t*

*<sup>b</sup>*(*r*)*dr*� − *<sup>τ</sup>*

�*γ*(*r*,*e*), *<sup>c</sup>*¯(*r*,*e*)�*π*(*de*)*dr*

 F*s t* (*ω*0)

tr{*σ*¯(*r*)�*P*(*t*)*σ*¯(*r*)}*dr* +

+ *o*(|*τ* − *t*|), as *τ* ↓ *t*, *a*.*e*.*t* ∈ [*s*, *T*).

It follows from (59), (60) and (61) that for any rational *τ* ∈ (*t*, *T*],

(*t*, *ω*0)) − *V*(*t*, *x*¯

*f*(*r*)*dr* + �*p*(*t*),

*s*,*y*;*u*¯

*τ* > *t*. By definition (30), then (52) holds. The proof is complete.

<sup>2</sup> tr

*s*,*y*;*u*¯

tr[*σ*¯(*r*)�*P*(*t*)*σ*¯(*r*)]*dr* +

As in (48) (using the duality technique), we have

$$\begin{split} & \quad V(\boldsymbol{\tau}, \boldsymbol{\tilde{x}}^{\sf s,\sf f:\mathcal{U}}(t,\omega\_{0})) - V(t, \boldsymbol{\tilde{x}}^{\sf s,\sf f:\mathcal{U}}(t,\omega\_{0})) \\ & \leq -\mathbb{E}\Big{(}-\int\_{t}^{\mathsf{T}} \boldsymbol{\tilde{f}}(\boldsymbol{r}) d\boldsymbol{r} \big| \boldsymbol{\mathcal{F}}\_{t}^{\sf s} \big{)}(\omega\_{0}) - \mathbb{E}\Big{(}\langle p(\boldsymbol{\tau}), \boldsymbol{\tilde{x}}\_{\sf s}(\boldsymbol{\tau})\rangle + \frac{1}{2} \boldsymbol{\tilde{\xi}}\_{\sf s}(\boldsymbol{\tau})^{\sf T} P(\boldsymbol{\tau}) \boldsymbol{\tilde{\xi}}\_{\sf s}(\boldsymbol{\tau}) \big{)} \mathcal{F}\_{t}^{\sf s} \big{)}(\omega\_{0}) \\ & + o(|\boldsymbol{\tau} - t|). \end{split} \tag{59}$$

Now let us estimate the terms on the right-hand side of (59). To this end, we first note that for any *<sup>ϕ</sup>*(·), *<sup>ϕ</sup>*ˆ(·) <sup>∈</sup> *<sup>L</sup>*<sup>2</sup> <sup>F</sup> ([0, *<sup>T</sup>*]; **<sup>R</sup>***n*), *<sup>ψ</sup>*(·) <sup>∈</sup> *<sup>L</sup>*<sup>2</sup> F,*p*([0, *<sup>T</sup>*]; **<sup>R</sup>***n*×*d*), <sup>Φ</sup>(·, ·)) <sup>∈</sup> *<sup>F</sup>*<sup>2</sup> *<sup>p</sup>* ([0, *T*]; **R***n*), we have the following three estimates:

**E** *<sup>τ</sup> t ϕ*(*r*)*dr*, *τ t ϕ*ˆ(*r*)*dr* F*s t* (*ω*0) ≤ **E** *τ t ϕ*(*r*)*dr* 2 F*s t* (*ω*0) 1 2 **E** *τ t ϕ*ˆ(*r*)*dr* 2 F*s t* (*ω*0) 1 2 ≤ (*τ* − *t*) *<sup>τ</sup> t* **E** |*ϕ*(*r*)| 2 F*s t* (*ω*0)*dr <sup>τ</sup> t* **E** |*ϕ*ˆ(*r*)| 2 F*s t* (*ω*0)*dr*<sup>1</sup> 2 = *o*(|*τ* − *t*|), as *τ* ↓ *t*, ∀*t* ∈ [*s*, *T*), **E** *<sup>τ</sup> t ϕ*(*r*)*dr*, *τ t ψ*(*r*)*dW*(*r*) F*s t* (*ω*0) ≤ **E** *τ t ϕ*(*r*)*dr* 2 F*s t* (*ω*0) 1 2 **E** *τ t ψ*(*r*)*dW*(*r*) 2 F*s t* (*ω*0) 1 2 ≤ (*τ* − *t*) 1 2 *<sup>τ</sup> t* **E** |*ϕ*(*r*)| 2 F*s t* (*ω*0)*dr <sup>τ</sup> t* **E** |*ψ*(*r*)| 2 F*s t* (*ω*0)*dr*<sup>1</sup> 2 = *o*(|*τ* − *t*|), as *τ* ↓ *t*, *a*.*e*.*t* ∈ [*s*, *T*), **E** *<sup>τ</sup> t ϕ*(*r*)*dr*, **E** *τ t* Φ(*r*,*e*)*N*˜ (*dedr*) F*s t* (*ω*0) ≤ **E** *τ t ϕ*(*r*)*dr* 2 F*s t* (*ω*0) 1 2 **E E** *τ t* Φ(*r*,*e*)*N*˜ (*dedr*) 2 F*s t* (*ω*0) 1 2 ≤ (*τ* − *t*) 1 2 *<sup>τ</sup> t* **E** |*ϕ*(*r*)| 2 F*s t* (*ω*0)*dr* **E** *τ t* **E** |Φ(*r*,*e*)| 2 F*s t* (*ω*0)*π*(*de*)*dr*<sup>1</sup> 2 = *o*(|*τ* − *t*|), as *τ* ↓ *t*, *a*.*e*.*t* ∈ [*s*, *T*).

All the last equalities in the above three inequalities is due to the fact that the sets of right Lebesgue points have full Lebesgue measures for integrable functions by Lemma 3.1 and *t* �→ F*s <sup>t</sup>* is right continuous in *t*. Thus applying Itô's formula to �*p*(·), *ξτ*(·)�, by (55), (18) we have

22 Will-be-set-by-IN-TECH

*ξτ*(*T*)�*hxx*(*x*¯

�*p*(*τ*), *ξτ*(*τ*)� +

Now let us estimate the terms on the right-hand side of (59). To this end, we first note that for

*f*(*r*)]*dr* + *h*(*xτ*(*T*)) − *h*(*x*¯

(*T*)), *ξτ*(*T*)�

(*T*))*ξτ*(*T*)

*ϕ*ˆ(*r*)*dr* 2 F*s t* (*ω*0) 1 2

*ψ*(*r*)*dW*(*r*)

Φ(*r*,*e*)*N*˜ (*dedr*)

2 F*s t* 

 2 F*s t* (*ω*0) 1 2

> 2 F*s t* (*ω*0) 1 2

 F*s t* 

*ξτ*(*τ*)�*P*(*τ*)*ξτ*(*τ*)

*s*,*y*;*u*¯

*s*,*y*;*u*¯

1 2

F,*p*([0, *<sup>T</sup>*]; **<sup>R</sup>***n*×*d*), <sup>Φ</sup>(·, ·)) <sup>∈</sup> *<sup>F</sup>*<sup>2</sup>

 *τ t* **E** |*ϕ*ˆ(*r*)| 2 F*s t* (*ω*0)*dr*

 *τ t* **E** |*ψ*(*r*)| 2 F*s t* (*ω*0)*dr*

*s*,*y*;*u*¯ (*T*)) F*s t* (*ω*0)

(*ω*0) + *o*(|*τ* − *t*|).

*<sup>p</sup>* ([0, *T*]; **R***n*), we have the

1 2

1 2

(*ω*0)*π*(*de*)*dr*

1 2 (59)

 F*s t* (*ω*0)

Let *ω*<sup>0</sup> ∈ Ω<sup>0</sup> be fixed, then for any rational *τ* > *t*, we have (noting (58))

(*t*, *ω*0))

[ *<sup>f</sup>*(*r*, *<sup>x</sup>τ*(*r*), *<sup>u</sup>*¯(*r*)) <sup>−</sup> ¯

*fx*(*r*), *ξτ*(*r*)�*dr* + �*hx*(*x*¯

1 2

(*t*, *ω*0))

*s*,*y*;*u*¯

*fxx*(*r*)*ξτ*(*r*)*dr* +

<sup>F</sup> ([0, *<sup>T</sup>*]; **<sup>R</sup>***n*), *<sup>ψ</sup>*(·) <sup>∈</sup> *<sup>L</sup>*<sup>2</sup>

 *τ t*

*ϕ*(*r*)*dr* 2 F*s t* (*ω*0) 1 2 **E** *τ t*

= *o*(|*τ* − *t*|), as *τ* ↓ *t*, ∀*t* ∈ [*s*, *T*),

= *o*(|*τ* − *t*|), as *τ* ↓ *t*, *a*.*e*.*t* ∈ [*s*, *T*),

Φ(*r*,*e*)*N*˜ (*dedr*)

*ψ*(*r*)*dW*(*r*)

 F*s t* (*ω*0)

> F*s t* (*ω*0)

 *<sup>τ</sup> t* **E** |*ϕ*(*r*)| 2 F*s t* (*ω*0)*dr*

 *τ t*

*ϕ*(*r*)*dr* 2 F*s t* (*ω*0) 1 2 **E** *τ t*

*ϕ*(*r*)*dr*,

*s*,*y*;*u*¯

(*ω*0) − **E**

*ϕ*ˆ(*r*)*dr* F*s t* (*ω*0)

*V*(*τ*, *x*¯

+ 1 2 *T τ*

*V*(*τ*, *x*¯

any *<sup>ϕ</sup>*(·), *<sup>ϕ</sup>*ˆ(·) <sup>∈</sup> *<sup>L</sup>*<sup>2</sup>

≤ − **E** − *τ t* ¯ *f*(*r*)*dr* F*s t* 

*s*,*y*;*u*¯

+ *o*(|*τ* − *t*|).

following three estimates:

**E** *<sup>τ</sup> t*

**E** *<sup>τ</sup> t*

**E** *<sup>τ</sup> t*

> ≤ **E** *τ t*

≤ (*τ* − *t*)

≤ **E** *τ t*

≤ (*τ* − *t*)

*ϕ*(*r*)*dr*, **E** *τ t*

> 1 2 *<sup>τ</sup> t* **E** |*ϕ*(*r*)| 2 F*s t* (*ω*0)*dr* **E** *τ t* **E** |Φ(*r*,*e*)|

≤ **E** *τ t*

≤ (*τ* − *t*)

*ϕ*(*r*)*dr*,

1 2 *<sup>τ</sup> t* **E** |*ϕ*(*r*)| 2 F*s t* (*ω*0)*dr*

*ϕ*(*r*)*dr* 2 F*s t* (*ω*0) 1 2 **E E** *τ t*

= *o*(|*τ* − *t*|), as *τ* ↓ *t*, *a*.*e*.*t* ∈ [*s*, *T*).

≤ **E** − *τ t* ¯ *f*(*r*)*dr* +

= **E** − *τ t* ¯ *f*(*r*)*dr* +

*s*,*y*;*u*¯

(*t*, *ω*0)) − *V*(*t*, *x*¯

*ξτ*(*r*)� ¯

 *T τ*

 *T τ* � ¯

As in (48) (using the duality technique), we have

(*t*, *ω*0)) − *V*(*t*, *x*¯

**E** �*p*(*τ*), *ξτ*(*τ*)� F*s t* (*ω*0) = **E** �*p*(*t*), *ξτ*(*τ*)� + �*p*(*τ*) − *p*(*t*), *ξτ*(*τ*)� F*s t* (*ω*0) = **E** *<sup>p</sup>*(*t*), <sup>−</sup> *τ t* ¯ *b*(*r*)*dr* − *τ t σ*¯(*r*)*dW*(*r*) − **E** *τ t c*¯(*r*,*e*)*N*˜ (*dedr*) + − *τ t* ¯ *bx*(*r*)*p*(*r*) + *<sup>σ</sup>*¯*x*(*r*)*q*(*r*) + **E** *c*¯*x*(*r*,*e*)*γ*(*r*,*e*)*π*(*de*) *dr* + *τ t <sup>q</sup>*(*r*)*dW*(*r*) + **E** *τ t <sup>γ</sup>*(*r*,*e*)*N*˜ (*dedr*), <sup>−</sup> *τ t* ¯ *b*(*r*)*dr* − *τ t σ*¯(*r*)*dW*(*r*) − **E** *τ t c*¯(*r*,*e*)*N*˜ (*dedr*) F*s t* (*ω*0) = **E** − �*p*(*t*), *τ t* ¯ *<sup>b</sup>*(*r*)*dr*� − *<sup>τ</sup> t* tr{*q*(*r*)�*σ*¯(*r*)}*dr* − **E** *τ t* �*γ*(*r*,*e*), *<sup>c</sup>*¯(*r*,*e*)�*π*(*de*)*dr* F*s t* (*ω*0) + *o*(|*τ* − *t*|), as *τ* ↓ *t*, *a*.*e*.*t* ∈ [*s*, *T*). (60)

Similarly, applying Itô's formula to *ξτ*(·)�*P*(·)*ξτ*(·)�, by (56) and (19), we have

$$\begin{split} & \mathbb{E} \left\{ \mathbb{E}\_{\boldsymbol{\tau}} (\boldsymbol{\tau})^{\top} P(\boldsymbol{\tau}) \boldsymbol{\xi}\_{\boldsymbol{\tau}} (\boldsymbol{\tau}) \Big| \mathcal{F}\_{l}^{\boldsymbol{s}} \right\} (\omega\_{0}) \\ &= \mathbb{E} \left\{ \int\_{t}^{\boldsymbol{\tau}} \text{tr} \{ \boldsymbol{\tau}(\boldsymbol{r})^{\top} P(t) \boldsymbol{\sigma}(\boldsymbol{r}) \} d\boldsymbol{r} + \int\_{\mathcal{E}} \int\_{t}^{\boldsymbol{\tau}} \text{tr} \{ \boldsymbol{\tilde{\boldsymbol{\tau}}}(\boldsymbol{r}, \boldsymbol{e})^{\top} P(t) \boldsymbol{\tilde{\boldsymbol{\tau}}}(\boldsymbol{r}, \boldsymbol{e}) \} \boldsymbol{\pi}(d\boldsymbol{e}) d\boldsymbol{r} \Big| \mathcal{F}\_{l}^{\boldsymbol{s}} \right\} (\omega\_{0}) \\ & \quad + o(|\boldsymbol{\tau} - t|), \quad \text{ as } \boldsymbol{\tau} \downarrow t, \quad a.e.t \in [\boldsymbol{s}\_{\boldsymbol{\tau}} T]. \end{split} \tag{61}$$

It follows from (59), (60) and (61) that for any rational *τ* ∈ (*t*, *T*],

*V*(*τ*, *x*¯ *s*,*y*;*u*¯ (*t*, *ω*0)) − *V*(*t*, *x*¯ *s*,*y*;*u*¯ (*t*, *ω*0)) ≤ **E** − *τ t* ¯ *f*(*r*)*dr* + �*p*(*t*), *τ t* ¯ *b*(*r*)*dr*� + *τ t* tr{*q*(*r*)�*σ*¯(*r*)}*dr* − 1 2 *τ t* tr[*σ*¯(*r*)�*P*(*t*)*σ*¯(*r*)]*dr* + **E** *τ t* �*γ*(*r*,*e*)�*c*¯(*r*,*e*)�*π*(*de*)*dr* − 1 2 **E** *τ t* tr{*c*¯(*r*,*e*)�*P*(*t*)*c*¯(*r*,*e*)}*π*(*de*)*dr* F*s t* (*ω*0) + *o*(|*τ* − *t*|) ≤ (*τ* − *t*) H(*t*, *x*¯ *s*,*y*;*u*¯ (*t*, *<sup>ω</sup>*0), *<sup>u</sup>*¯(*t*)) + <sup>1</sup> 2 tr *R*(*t*,*e*)*c*¯(*t*,*e*)*c*¯(*t*,*e*)� *π*(*de*) = (*τ* − *t*)G(*t*, *x*¯ *s*,*y*;*u*¯ (*t*, *ω*0), *u*¯(*t*)) + *o*(|*τ* − *t*|), as *τ* ↓ *t*, *a*.*e*.*t* ∈ [*s*, *T*). (62)

By the same argument as in the paragraph following (49), we conclude that (62) holds for any *τ* > *t*. By definition (30), then (52) holds. The proof is complete.

**Remark 3.4** As aforementioned, it is worth to point out that by comparing the result of Theorem 3.3 with those analogue in the diffusion case (see Theorem 4.7 on pp. 263 of [18]), we observe that an additional term – <sup>1</sup> <sup>2</sup> tr *R*(*t*,*e*)*c*¯(*t*,*e*)*c*¯(*t*,*e*)� *π*(*de*) – containing the integrand with respect to the compensated martingale measure *N*˜ in the second-order adjoint equation appears in the G-function of (52). This is different from the continuous diffusion case where this G-function coincides with the H-function appearing in the maximum principle.

It is not difficulty to verify that the following function is a viscosity solution of (3.2):

we have (noting (18))

We can solve that

Then with

⎧ ⎪⎪⎪⎪⎪⎨ P1,2,<sup>+</sup> *<sup>t</sup>*+,*<sup>x</sup> V*(*t*, *x*¯

G(*t*, *x*¯

<sup>P</sup>1,2,<sup>−</sup> *<sup>t</sup>*+,*<sup>x</sup> <sup>V</sup>*(*t*, *<sup>x</sup>*¯

0,0;*u*¯

⎪⎪⎪⎪⎪⎩

[0, *T*).

(*x*¯*s*,*y*;*u*¯

*Vt*(*t*, *x*¯

*s*,*y*;*u*¯

(*t*)) = *G*

� *t*, *x*¯ *s*,*y*;*u*¯

= sup *u*∈**U** *G* � *t*, *x*¯ *s*,*y*;*u*¯

*p*(*T*) = 1.

H(*t*, *x*¯

*s*,*y*;*u*¯

by Theorem 2.3, we have for *t* ∈ [*s*, *T*],

Now consider *s* = 0, *y* = 0. Clearly, (*x*¯0,0;*u*¯

[0, *T*]. However, one can show that

0,0;*u*¯

0,0;*u*¯

(*t*)) = <sup>P</sup>1,2,<sup>+</sup>

*p*(*t*) = 1, *t* ∈ [0, *T*], *P*(*t*) = 0, *t* ∈ [0, *T*],

(*t*), *u*¯(*t*)) = G(*t*, 0, 0) = 0, *t* ∈ [0, *T*].

(*t*), *u*) = *e*

� *<sup>T</sup> <sup>t</sup> <sup>u</sup>*¯(*s*)*dsx*¯

*<sup>V</sup>*(*t*, *<sup>x</sup>*) = � <sup>−</sup> *<sup>x</sup>*, if *<sup>x</sup>* <sup>≤</sup> 0, <sup>−</sup> *xeT*−*<sup>t</sup>*

which clearly satisfy (12) and (13). Thus, by the uniqueness of the viscosity solution, *V* coincides with the value function of the optimal control problem. However, it is *not* in *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>**), since *Vx*(*t*, *<sup>x</sup>*) has a jump at (*t*, 0) for all 0 <sup>≤</sup> *<sup>t</sup>* <sup>&</sup>lt; *<sup>T</sup>*. On the other hand,

� <sup>−</sup>*dp*(*t*)=[*u*¯(*t*)*p*(*t*) + *<sup>q</sup>*(*t*) + *λγ*(*t*)*u*¯(*t*)]*dt* <sup>−</sup> *<sup>q</sup>*(*t*)*dW*(*t*) <sup>−</sup> *<sup>γ</sup>*(*t*)*N*˜ (*dt*), *<sup>t</sup>* <sup>∈</sup> [*s*, *<sup>T</sup>*],

� *<sup>T</sup>*

(*t*)*u* − *λu*¯(*t*)*x*¯

*s*,*y*;*u*¯

*<sup>t</sup> <sup>u</sup>*¯(*s*)*ds*, 0, 0).

*s*,*y*;*u*¯

(*t*) > 0,

(*t*) ≤ 0.

(*t*)2*u* +

1 2 *λx*¯ *s*,*y*;*u*¯

(·), *u*¯(·)) ≡ (0, 0) is an optimal control. Theorem

, −1] × [0, ∞), *t* ∈ [0, *T*],

(*t*)), −*Vxx*(*t*, *x*¯

(*t*)), −*Vxx*(*t*, *x*¯

*a*.*e*.*t* ∈ [*s*, *T*], **P***-a*.*s*.,

*s*,*y*;*u*¯ (*t*))�

> *s*,*y*;*u*¯ (*t*))� ,

(*t*)2*u*2,

Stochastic Control for Jump Diff usions 143

(*t*), *t* ∈

(65)

(66)

(*p*(*t*), *q*(*t*), *γ*(*t*)) = (*e*

*s*,*y*;*u*¯

*<sup>u</sup>*¯(*t*) = � 1, *<sup>x</sup>*¯

0, *x*¯ *s*,*y*;*u*¯

*<sup>t</sup>*+,*<sup>x</sup> <sup>V</sup>*(*t*, 0)=[0, <sup>∞</sup>) <sup>×</sup> [−*eT*−*<sup>t</sup>*

(*t*)) = <sup>P</sup>1,2,<sup>−</sup> *<sup>t</sup>*+,*<sup>x</sup> <sup>V</sup>*(*t*, 0)=[−∞, 0] <sup>×</sup> <sup>∅</sup> <sup>×</sup> <sup>∅</sup>, *<sup>t</sup>* <sup>∈</sup> [0, *<sup>T</sup>*],

Thus, it is clear that both the set inclusions in (63) and (64) of Theorem 3.4 are strict for *t* ∈

The following result is the special case when we assume the value function is enough smooth. **Corollary 3.1** *Suppose that* **(H1)**∼**(H3)** *hold and let* (*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> be fixed. Let*

*be the first-order adjoint processes. Suppose the value function V* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*. Then*

(*t*), *u*¯(*t*), −*V*(*t*, *x*¯

(*t*), *u*, −*V*(*t*, *x*¯

(·), *u*¯(·)) *be an optimal pair of our stochastic optimal control problem and* (*p*(·), *q*(·), *γ*(·, ·))

(*t*)), −*Vx*(*t*, *x*¯

(*t*)), −*Vx*(*t*, *x*¯

*s*,*y*;*u*¯

*s*,*y*;*u*¯

*s*,*y*;*u*¯

*s*,*y*;*u*¯

2.1 of [7] does not apply, because *Vx*(*t*, *x*) does not exist along the *whole* trajectory *x*¯0,0;*u*¯

, if *x* > 0,

Now, let us combine Theorem 3.2 and 3.3 to get the following result.

**Theorem 3.4** *Suppose that* **(H1)**∼**(H3)** *hold and let* (*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> be fixed. Let* (*x*¯*s*,*y*;*u*¯ (·), *u*¯(·)) *be the optimal pair of our stochastic control problem. Let* (*p*(·), *q*(·), *γ*(·, ·)) *and* (*P*(·), *Q*(·), *R*(·, ·)) *are first-order and second-order adjoint processes, respectively. Then we have*

$$\begin{aligned} \left[ \mathcal{G}(t, \bar{\mathfrak{x}}^{\circ \circ \mathfrak{X}}(t), \bar{\mathfrak{u}}(t)), \infty \right) \times \{ -p(t) \} \times [-P(t), \infty) &\subseteq \mathcal{P}^{1, 2, +}\_{t + \mathcal{X}} V(t, \bar{\mathfrak{x}}^{\circ \circ \mathfrak{X}}(t)), \ a.e. t \in [\mathrm{s}, T], \mathsf{P}\text{-a.s.} \\ \left[ \mathcal{P}^{1, 2, -}\_{t + \mathcal{X}} V(t, \bar{\mathfrak{x}}^{\circ \circ \mathfrak{X}}(t)) \subseteq (-\infty, \mathcal{G}(t, \bar{\mathfrak{x}}^{\circ \circ \mathfrak{X}}(t), \bar{\mathfrak{u}}(t))] \times \{ -p(t) \} \times (-\infty, P(t)], \ a.e. t \in [\mathrm{s}, T], \mathsf{P}\text{-a.s.} \end{aligned} \tag{63}$$

*Proof* The first conclusion (63) can be proved by combining the proofs of (34) and (52) and making use of (3.1). We now show (64). For any *<sup>q</sup>* ∈ P1,<sup>−</sup> *<sup>t</sup>*<sup>+</sup> *<sup>V</sup>*(*t*, *<sup>x</sup>*¯*s*,*y*;*u*¯ (*t*)), by definition (31) and (62) we have

$$\begin{aligned} 0 \le & V(\boldsymbol{\tau}, \boldsymbol{\bar{x}}^{\boldsymbol{s}, \boldsymbol{y}; \boldsymbol{\mathcal{R}}}(t)) - V(t, \boldsymbol{\bar{x}}^{\boldsymbol{s}, \boldsymbol{y}; \boldsymbol{\mathcal{R}}}(t)) - q(\boldsymbol{\tau} - t) \\ &= (\mathcal{G}(t, \boldsymbol{\bar{x}}^{\boldsymbol{s}, \boldsymbol{y}; \boldsymbol{\mathcal{R}}}(t), \boldsymbol{\bar{u}}(t)) - q)(\boldsymbol{\tau} - t) + o(|\boldsymbol{\tau} - t|), \quad a.e. t \in [s, T], \boldsymbol{\mathsf{P}}\text{-a.s.} \end{aligned}$$

Then it is necessary that

$$q \le \mathcal{G}(t, \mathfrak{x}^{\mathbf{s}, \mathbf{y}; \mathcal{U}}(t), \mathfrak{u}(t)), \quad a.e. t \in [\mathbf{s}, T], \mathbf{P}\text{-}a.\text{s.t}$$

From this and (35), we have (64). The proof is complete.

Theorem 3.4 is a generalization of the classical result on the relationship between stochastic MP and DPP (see Theorem 2.1 of [7]). On the other hand, we do have a simple example showing that both the set inclusions in (63) and (64) may be strict.

**Example 3.2** Consider the following linear stochastic control system with Poisson jumps (*n* = *d* = 1):

$$\begin{cases} d\mathbf{x}^{\mathbf{s},\mathbf{y};\mu}(t) = \mathbf{x}^{\mathbf{s},\mathbf{y};\mu}(t)u(t)dt + \mathbf{x}^{\mathbf{s},\mathbf{y};\mu}(t)dW(t) + \mathbf{x}^{\mathbf{s},\mathbf{y};\mu}(t-)u(t)\tilde{N}(dt), \quad t \in [\mathbf{s},T],\\ \mathbf{x}^{\mathbf{s},\mathbf{y};\mu}(\mathbf{s}) = \mathbf{y}. \end{cases}$$

Here *<sup>N</sup>* is a Poisson process with the intensity *<sup>λ</sup>dt* and *<sup>N</sup>*˜ (*dt*) :<sup>=</sup> *<sup>N</sup>*(*dt*) <sup>−</sup> *<sup>λ</sup>dt*(*<sup>λ</sup>* <sup>&</sup>gt; <sup>0</sup>) is the compensated martingale measure. The control domain is **U** = [0, 1] and the cost functional is

$$J(\mathbf{s}, \mathbf{y}; \mathfrak{u}(\cdot)) = -\mathbb{E} \mathfrak{x}^{\mathbf{s}, \mathbf{y}; \mathfrak{u}}(T).$$

The corresponding HJB equation (15) now reads

$$\begin{cases} -\upsilon\_t(t, \mathbf{x}) - \lambda \upsilon(t, \mathbf{x}) - \frac{1}{2} \mathbf{x}^2 \upsilon\_{\mathbf{xx}}(t, \mathbf{x}) \\ \quad + \sup\_{0 \le \underline{u} \le 1} \left[ (1 - \lambda) \upsilon\_{\mathbf{x}}(t, \mathbf{x}) \mathbf{x} \boldsymbol{\mu} + \lambda \upsilon(t, \mathbf{x} + \mathbf{x} \boldsymbol{\mu}) \right] = 0, \quad t \in [\mathsf{s}, T]\_{\mathsf{s}} \\ \upsilon(T, \mathbf{x}) = -\mathsf{x}. \end{cases}$$

It is not difficulty to verify that the following function is a viscosity solution of (3.2):

$$V(t,x) = \begin{cases} -\infty, & \text{if } x \le 0, \\\ -\infty e^{T-t}, & \text{if } x > 0, \end{cases}$$

which clearly satisfy (12) and (13). Thus, by the uniqueness of the viscosity solution, *V* coincides with the value function of the optimal control problem. However, it is *not* in *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>**), since *Vx*(*t*, *<sup>x</sup>*) has a jump at (*t*, 0) for all 0 <sup>≤</sup> *<sup>t</sup>* <sup>&</sup>lt; *<sup>T</sup>*. On the other hand, we have (noting (18))

$$\begin{cases} -dp(t) = [\overline{u}(t)p(t) + q(t) + \lambda \gamma(t)\overline{u}(t)]dt - q(t)dW(t) - \gamma(t)\tilde{N}(dt), \quad t \in [s, T],\\ p(T) = 1. \end{cases}$$

We can solve that

$$(p(t), q(t), \gamma(t)) = (e^{\int\_t^T R(s)ds}, 0, 0).$$

Then with

24 Will-be-set-by-IN-TECH

with respect to the compensated martingale measure *N*˜ in the second-order adjoint equation appears in the G-function of (52). This is different from the continuous diffusion case where

**Theorem 3.4** *Suppose that* **(H1)**∼**(H3)** *hold and let* (*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> be fixed. Let*

*Proof* The first conclusion (63) can be proved by combining the proofs of (34) and (52) and

Theorem 3.4 is a generalization of the classical result on the relationship between stochastic MP and DPP (see Theorem 2.1 of [7]). On the other hand, we do have a simple example

**Example 3.2** Consider the following linear stochastic control system with Poisson jumps

� *dxs*,*y*;*u*(*t*) = *<sup>x</sup>s*,*y*;*u*(*t*)*u*(*t*)*dt* <sup>+</sup> *<sup>x</sup>s*,*y*;*u*(*t*)*dW*(*t*) + *<sup>x</sup>s*,*y*;*u*(*t*−)*u*(*t*)*N*˜ (*dt*), *<sup>t</sup>* <sup>∈</sup> [*s*, *<sup>T</sup>*],

Here *<sup>N</sup>* is a Poisson process with the intensity *<sup>λ</sup>dt* and *<sup>N</sup>*˜ (*dt*) :<sup>=</sup> *<sup>N</sup>*(*dt*) <sup>−</sup> *<sup>λ</sup>dt*(*<sup>λ</sup>* <sup>&</sup>gt; <sup>0</sup>) is the compensated martingale measure. The control domain is **U** = [0, 1] and the cost functional is

*<sup>J</sup>*(*s*, *<sup>y</sup>*; *<sup>u</sup>*(·)) = <sup>−</sup>**E***xs*,*y*;*u*(*T*).

*x*2*vxx*(*t*, *x*)

[(1 − *λ*)*vx*(*t*, *x*)*xu* + *λv*(*t*, *x* + *xu*)] = 0, *t* ∈ [*s*, *T*],

2

(*t*)) − *q*(*τ* − *t*)

(*t*), *u*¯(*t*)), *a*.*e*.*t* ∈ [*s*, *T*], **P**-*a*.*s*.

(*P*(·), *Q*(·), *R*(·, ·)) *are first-order and second-order adjoint processes, respectively. Then we have*

(·), *u*¯(·)) *be the optimal pair of our stochastic control problem. Let* (*p*(·), *q*(·), *γ*(·, ·)) *and*

*<sup>t</sup>*+,*<sup>x</sup> V*(*t*, *x*¯

*<sup>t</sup>*<sup>+</sup> *<sup>V</sup>*(*t*, *<sup>x</sup>*¯*s*,*y*;*u*¯

(*τ* − *t*) + *o*(|*τ* − *t*|), *a*.*e*.*t* ∈ [*s*, *T*], **P**-*a*.*s*.

*s*,*y*;*u*¯

(*t*), *u*¯(*t*))] × {−*p*(*t*)} × (−∞, *P*(*t*)], *a*.*e*.*t* ∈ [*s*, *T*], **P***-a*.*s*.

(*t*)), *a*.*e*.*t* ∈ [*s*, *T*], **P***-a*.*s*.,

(*t*)), by definition (31) and

(63)

(64)

this G-function coincides with the H-function appearing in the maximum principle.

Now, let us combine Theorem 3.2 and 3.3 to get the following result.

(*t*), *<sup>u</sup>*¯(*t*)), <sup>∞</sup>) × {−*p*(*t*)} × [−*P*(*t*), <sup>∞</sup>) ⊆ P1,2,<sup>+</sup>

*s*,*y*;*u*¯

*s*,*y*;*u*¯

�

(*t*)) ⊆ (−∞, G(*t*, *x*¯

*s*,*y*;*u*¯

making use of (3.1). We now show (64). For any *<sup>q</sup>* ∈ P1,<sup>−</sup>

*q* ≤ G(*t*, *x*¯

From this and (35), we have (64). The proof is complete.

showing that both the set inclusions in (63) and (64) may be strict.

(*t*)) − *V*(*t*, *x*¯

(*t*), *u*¯(*t*)) − *q*

*s*,*y*;*u*¯

(*x*¯*s*,*y*;*u*¯

[G(*t*, *x*¯

<sup>P</sup>1,2,<sup>−</sup> *<sup>t</sup>*+,*<sup>x</sup> <sup>V</sup>*(*t*, *<sup>x</sup>*¯

(62) we have

(*n* = *d* = 1):

*s*,*y*;*u*¯

*s*,*y*;*u*¯

0 ≤ *V*(*τ*, *x*¯

= � G(*t*, *x*¯ *s*,*y*;*u*¯

*xs*,*y*;*u*(*s*) = *y*.

⎧ ⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

The corresponding HJB equation (15) now reads

+ sup 0≤*u*≤1

*v*(*T*, *x*) = −*x*.

<sup>−</sup> *vt*(*t*, *<sup>x</sup>*) <sup>−</sup> *<sup>λ</sup>v*(*t*, *<sup>x</sup>*) <sup>−</sup> <sup>1</sup>

Then it is necessary that

$$\mathcal{H}(t, \mathfrak{x}^{\mathfrak{s}, \mathfrak{y}; \mathfrak{A}}(t), u) = e^{\int\_{t}^{\mathfrak{T}} \mathfrak{a}(\mathfrak{s}) d\mathfrak{s}} \mathfrak{x}^{\mathfrak{s}, \mathfrak{y}; \mathfrak{A}}(t) u - \lambda \mathfrak{a}(t) \mathfrak{x}^{\mathfrak{s}, \mathfrak{y}; \mathfrak{A}}(t)^{2} u + \frac{1}{2} \lambda \mathfrak{x}^{\mathfrak{s}, \mathfrak{y}; \mathfrak{A}}(t)^{2} u^{2} \lambda$$

by Theorem 2.3, we have for *t* ∈ [*s*, *T*],

$$\mathfrak{u}(t) = \begin{cases} 1, & \mathfrak{x}^{s,y;\mathfrak{a}}(t) > 0, \\ 0, & \mathfrak{x}^{s,y;\mathfrak{a}}(t) \le 0. \end{cases}$$

Now consider *s* = 0, *y* = 0. Clearly, (*x*¯0,0;*u*¯ (·), *u*¯(·)) ≡ (0, 0) is an optimal control. Theorem 2.1 of [7] does not apply, because *Vx*(*t*, *x*) does not exist along the *whole* trajectory *x*¯0,0;*u*¯ (*t*), *t* ∈ [0, *T*]. However, one can show that

$$\begin{cases} \mathcal{P}^{1,2,+}\_{t+\mathcal{X}}V(t, \mathfrak{x}^{0,0;t}(t)) = \mathcal{P}^{1,2,+}\_{t+\mathcal{X}}V(t,0) = [0,\infty) \times [-e^{T-t}\_{-}, -1] \times [0,\infty), \quad t \in [0,T], \\\mathcal{P}^{1,2,-}\_{t+\mathcal{X}}V(t, \mathfrak{x}^{0,0;t}(t)) = \mathcal{P}^{1,2,-}\_{t+\mathcal{X}}V(t,0) = [-\infty,0] \times \mathcal{Q} \times \mathcal{Q}, \quad t \in [0,T], \\\ p(t) = 1, \quad t \in [0,T], \qquad P(t) = 0, \quad t \in [0,T], \\\ \mathcal{G}(t, \mathfrak{x}^{0,0;t}(t), \mathfrak{u}(t)) = \mathcal{G}(t,0,0) = 0, \quad t \in [0,T]. \end{cases} \tag{65}$$

Thus, it is clear that both the set inclusions in (63) and (64) of Theorem 3.4 are strict for *t* ∈ [0, *T*).

The following result is the special case when we assume the value function is enough smooth.

**Corollary 3.1** *Suppose that* **(H1)**∼**(H3)** *hold and let* (*s*, *<sup>y</sup>*) <sup>∈</sup> [0, *<sup>T</sup>*) <sup>×</sup> **<sup>R</sup>***<sup>n</sup> be fixed. Let* (*x*¯*s*,*y*;*u*¯ (·), *u*¯(·)) *be an optimal pair of our stochastic optimal control problem and* (*p*(·), *q*(·), *γ*(·, ·)) *be the first-order adjoint processes. Suppose the value function V* <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*. Then*

*Vt*(*t*, *x*¯ *s*,*y*;*u*¯ (*t*)) = *G* � *t*, *x*¯ *s*,*y*;*u*¯ (*t*), *u*¯(*t*), −*V*(*t*, *x*¯ *s*,*y*;*u*¯ (*t*)), −*Vx*(*t*, *x*¯ *s*,*y*;*u*¯ (*t*)), −*Vxx*(*t*, *x*¯ *s*,*y*;*u*¯ (*t*))� = sup *u*∈**U** *G* � *t*, *x*¯ *s*,*y*;*u*¯ (*t*), *u*, −*V*(*t*, *x*¯ *s*,*y*;*u*¯ (*t*)), −*Vx*(*t*, *x*¯ *s*,*y*;*u*¯ (*t*)), −*Vxx*(*t*, *x*¯ *s*,*y*;*u*¯ (*t*))� , *a*.*e*.*t* ∈ [*s*, *T*], **P***-a*.*s*.,

(66)

*where G is defined by (16). Moreover, if V* <sup>∈</sup> *<sup>C</sup>*1,3([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *and Vtx is also continuous, then*

$$\begin{cases} V\_{\mathbf{x}}(t, \widetilde{\mathbf{x}}^{\sf s,\mathbf{y},\mathbf{z}}(t)) = -p(t), & \forall t \in [\mathbf{s}, T], \mathbf{P}\text{-a.s.}, \\\\ V\_{\mathbf{x}\mathbf{x}}(t, \widetilde{\mathbf{x}}^{\sf s,\mathbf{y},\mathbf{z}}(t))\widetilde{\boldsymbol{\sigma}}(t) = -q(t), & a.e.t \in [\mathbf{s}, T], \mathbf{P}\text{-a.s.}, \\\\ V\_{\mathbf{x}}(t, \widetilde{\mathbf{x}}^{\sf s,\mathbf{y},\mathbf{z}}(t-) + \widetilde{\boldsymbol{\varepsilon}}(t, \cdot)) - V\_{\mathbf{x}}(t, \widetilde{\mathbf{x}}^{\sf s,\mathbf{y},\mathbf{z}}(t-)) = -\gamma(t, \cdot), & a.e.t \in [\mathbf{s}, T], \mathbf{P}\text{-a.s.} \end{cases} \tag{67}$$

*Proof* By (63) and the fact that *V* is a viscosity solution of (15), we have

*u*∈**U**

(*t*), *u*, −*ψ*1(*t*, *x*¯

*G*(*t*, *x*¯ *s*,*y*;*u*¯

*s*,*y*;*u*¯

+ 1 2 tr *P*(*t*) **E**

*ψ*1(*t*, *x* + *c*¯(*t*,*e*)) − *ψ*1(*t*, *x*) + �*p*(*t*) + *γ*(*t*,*e*), *c*¯(*t*,*e*)�

In this chapter, we have derived the relationship between the maximum principle and dynamic programming principle for the stochastic optimal control problem of jump diffusions. Without involving any derivatives of the value function, relations among the adjoint processes, the generalized Hamiltonian and the value function are derived in the language of viscosity solutions and the associated super-subjets. The conditions under which the above results are valid are very mild and reasonable. The results in this chapter bridge an

[1] Alvarez, D. & Tourin, A. (1996). Viscosity solutions of nonlinear integral-differential

[2] Barles, G., Buckdahn, R. & Pardoux, E. (1997). Backward stochastic differential equations and integral-partial differential equations. *Stochastics & Stochastics Reports*,

[3] Barles, G. & Imbert, C. (2008). Second-order elliptic integral-differential equations: Viscosity solutions' theory revisited. *Ann. Inst. H. Poincaré Anal. Non Linéaire*, Vol. 25,

[4] Bensoussan, A. (1981). Lectures on stochastic control. *Lecture Notes in Mathematics*, Vol.

[5] Bismut, J.M. (1978). An introductory approach to duality in optimal stochastic control.

[6] Biswas, I.H., Jakobsen, E.R. & Karlsen, K.H. (2010). Viscosity solutions for a system of integro-PDEs and connections to optimal switching and control of jump-diffusion

*SIAM Journal on Control and Optimization*, Vol. 20, No. 1, 62–78.

processes. *Applied Mathematics and Optimization*, Vol. 62, No. 1, 47–80.

equations. *Ann. Inst. H. Poincaré Anal. Non Linéaire*, Vol. 13, No. 3, 293–317.

*s*,*y*;*u*¯

(*t*), *u*, −*ψ*1(*t*, *x*¯

(*t*)), *p*(*t*), *P*(*t*))

(*t*)), *p*(*t*), *P*(*t*))

*s*,*y*;*u*¯

*c*¯(*t*,*e*)*c*¯(*t*,*e*)�*π*(*de*)

 *π*(*de*).

(*t*)), *p*(*t*), *P*(*t*))

Stochastic Control for Jump Diff usions 145

(*t*), *u*¯(*t*)) + sup

(*t*), *u*¯(*t*), −*ψ*1(*t*, *x*¯

*<sup>σ</sup>*¯(*t*)�[*q*(*t*) <sup>−</sup> *<sup>P</sup>*(*t*)*σ*¯(*t*)]

0 ≥−G(*t*, *x*¯

≥ − *G*(*t*, *x*¯

+ sup *u*∈**U**

<sup>−</sup> tr

− **E** 

important gap in the literature.

*Shandong University, P. R. China*

Vol. 60, 57–83.

No. 3, 567–585.

972, Springer–Verlag, Berlin.

**4. Conclusion**

**Author details**

**5. References**

Jingtao Shi

*s*,*y*;*u*¯

*s*,*y*;*u*¯

*G*(*t*, *x*¯ *s*,*y*;*u*¯

Then, (68) or (69) follows. The proof is complete.

By martingale representation theorem (see Lemma 2.3, [16]) and Itô's formula (see [10]), the proof technique is quite similar to Theorem 4.1, Chapter 4 of [18]. So we omit the detail. In fact, the relationship in (67) also can be seen in Theorem 2.1 of [7]. See also (3) in Introduction of this chapter.

**Remark 3.5** (i) On the assumption that the value function *V* is smooth, the first equality in (66) show us the relationship between the derivative of *V* with respective to the time variable and the generalized Hamiltonian function *G* defined by (16) along an optimal state trajectory.

(ii) It is interesting to note that the second equality in (66) may be regard as a "maximum principle" in terms of the value function and its derivatives. It is different from the stochastic MP aforementioned (Theorem 2.3), where no value function or its derivatives is involved.

(iii) The three equalities in (67) show us the relationship between the derivative of *V* with respective to the state variable and the adjoint processes *p*, *q*, *γ*(·). More precisely, the three adjoint processes *p*, *q*, *γ*(·) can be expressed in terms of the derivatives of *V* with respective to the state variable along an optimal state trajectory. It is also interesting to note that from the third equality in (67), we observe that the jump amplitude of *Vx*(*t*, *x*¯*s*,*y*;*u*¯ (*t*)) equal to −*γ*(*t*, ·) which is just that *Vx*(*t*, *x*¯*s*,*y*;*u*¯ (*t*)) = −*p*(*t*) tell us by the first-order adjoint equation (18).

**Remark 3.6** By Remark 3.2 and Example 3.1, it can be seen that though the first classical relation in (67) of Corollary 3.1 is recovered from Theorem 3.2 when the value function *V* is smooth enough, the nonsmooth version of the second classical relation in (67), i.e., *q*(*t*) = *P*(*t*)*σ*¯(*t*), does not hold in general. We are also interested to the question that to what extent the third classical relation in (67) can be generalized when *V* is not smooth. However, it seems that Theorem 3.2 tells us nothing in this context while the following result gives the general relationship among *p*, *q*, *γ*(·), *P*, *σ*¯ and *c*¯(·).

**Proposition 3.3** *Under the assumption of Theorem 3.4, we have*

$$\begin{split} 0 \le & \operatorname{tr} \Big\{ \bar{\boldsymbol{\sigma}}(t)^{\top} \left[ q(t) - P(t) \bar{\boldsymbol{\sigma}}(t) \right] \} - \frac{1}{2} \operatorname{tr} \Big\{ P(t) \int\_{\mathbf{E}} \bar{\boldsymbol{\varepsilon}}(t, e) \bar{\boldsymbol{\varepsilon}}(t, e)^{\top} \boldsymbol{\pi}(de) \Big\} \\ & + \int\_{\mathbf{E}} \Big[ \psi\_{1}(t, \mathbf{x} + \bar{\boldsymbol{\varepsilon}}(t, e)) - \psi\_{1}(t, \mathbf{x}) + \langle p(t) + \gamma(t, e), \bar{\boldsymbol{\varepsilon}}(t, e) \rangle \Big] \boldsymbol{\pi}(de), \quad a.e.t \in [s\_{\boldsymbol{\prime}}, T], \mathbf{P}\text{-a.s.} \end{split} \tag{68}$$

*or, equivalently,*

$$\mathcal{G}(t, \bar{\mathfrak{x}}^{\underline{s}, \underline{y}; \bar{\mu}}(t), \bar{u}(t)) \ge G(t, \bar{\mathfrak{x}}^{\underline{s}, \underline{y}; \bar{\mu}}(t), \bar{u}(t), -\psi\_1(t, \bar{\mathfrak{x}}^{\underline{s}, \underline{y}; \bar{\mu}}(t)), p(t), P(t)), \quad \text{a.e.} t \in [\mathsf{s}, T], \mathbb{P}\text{-a.s.}, \tag{69}$$
 
$$where \; \psi\_1 \in \mathbb{C}^{1,2}([0, T] \times \mathbb{R}^n), \text{ such that } \psi\_1(t', \mathfrak{x}') > V(t', \mathfrak{x}'), \forall (t', \mathfrak{x}') \ne (t, \mathfrak{x}) \in [\mathsf{s}, T] \times \mathbb{R}^n.$$

*Proof* By (63) and the fact that *V* is a viscosity solution of (15), we have

$$\begin{split} 0 &\geq -\mathcal{G}(t, \bar{\mathbf{x}}^{\bar{\mathbf{x}}, \mathcal{Y}; \mathbb{R}}(t), \bar{u}(t)) + \sup\_{\mathsf{u} \in \mathsf{U}} G(t, \bar{\mathbf{x}}^{\bar{\mathbf{x}}, \mathcal{Y}; \mathbb{R}}(t), \mathsf{u}\_{\prime} - \boldsymbol{\Psi}\_{1}(t, \bar{\mathbf{x}}^{\bar{\mathbf{x}}, \mathcal{Y}; \mathbb{R}}(t)), p(t), P(t)) \\ &\geq -\mathcal{G}(t, \bar{\mathbf{x}}^{\bar{\mathbf{x}}, \mathcal{Y}; \mathbb{R}}(t), \bar{u}(t), -\boldsymbol{\Psi}\_{1}(t, \bar{\mathbf{x}}^{\bar{\mathbf{x}}, \mathcal{Y}; \mathbb{R}}(t)), p(t), P(t)) \\ &+ \sup\_{\mathsf{u} \in \mathsf{U}} G(t, \bar{\mathbf{x}}^{\bar{\mathbf{x}}, \mathcal{Y}; \mathbb{R}}(t), \mathsf{u}\_{\prime} - \boldsymbol{\Psi}\_{1}(t, \bar{\mathbf{x}}^{\bar{\mathbf{x}}, \mathcal{Y}; \mathbb{R}}(t)), p(t), P(t)) \\ &- \operatorname{tr} \{ \bar{\sigma}(t)^{\top} [q(t) - P(t)\bar{\sigma}(t)] \} + \frac{1}{2} \operatorname{tr} \{ P(t) \int\_{\mathbf{E}} \bar{\varepsilon}(t, e) \bar{\varepsilon}(t, e)^{\top} \boldsymbol{\pi}(de) \} \\ &- \int\_{\mathbf{E}} \left[ \psi\_{1}(t, \mathsf{x} + \bar{\varepsilon}(t, e)) - \psi\_{1}(t, \mathsf{x}) + \langle p(t) + \gamma(t, e), \bar{\varepsilon}(t, e) \rangle \right] \boldsymbol{\pi}(de) . \end{split}$$

Then, (68) or (69) follows. The proof is complete.

## **4. Conclusion**

26 Will-be-set-by-IN-TECH

*where G is defined by (16). Moreover, if V* <sup>∈</sup> *<sup>C</sup>*1,3([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*) *and Vtx is also continuous, then*

(*t*)) = −*p*(*t*), ∀*t* ∈ [*s*, *T*], **P***-a*.*s*.,

third equality in (67), we observe that the jump amplitude of *Vx*(*t*, *x*¯*s*,*y*;*u*¯

(*t*−) + *c*¯(*t*, ·)) − *Vx*(*t*, *x*¯

(*t*))*σ*¯(*t*) = −*q*(*t*), *a*.*e*.*t* ∈ [*s*, *T*], **P***-a*.*s*.,

*s*,*y*;*u*¯

By martingale representation theorem (see Lemma 2.3, [16]) and Itô's formula (see [10]), the proof technique is quite similar to Theorem 4.1, Chapter 4 of [18]. So we omit the detail. In fact, the relationship in (67) also can be seen in Theorem 2.1 of [7]. See also (3) in Introduction

**Remark 3.5** (i) On the assumption that the value function *V* is smooth, the first equality in (66) show us the relationship between the derivative of *V* with respective to the time variable and the generalized Hamiltonian function *G* defined by (16) along an optimal state trajectory. (ii) It is interesting to note that the second equality in (66) may be regard as a "maximum principle" in terms of the value function and its derivatives. It is different from the stochastic MP aforementioned (Theorem 2.3), where no value function or its derivatives is involved. (iii) The three equalities in (67) show us the relationship between the derivative of *V* with respective to the state variable and the adjoint processes *p*, *q*, *γ*(·). More precisely, the three adjoint processes *p*, *q*, *γ*(·) can be expressed in terms of the derivatives of *V* with respective to the state variable along an optimal state trajectory. It is also interesting to note that from the

**Remark 3.6** By Remark 3.2 and Example 3.1, it can be seen that though the first classical relation in (67) of Corollary 3.1 is recovered from Theorem 3.2 when the value function *V* is smooth enough, the nonsmooth version of the second classical relation in (67), i.e., *q*(*t*) = *P*(*t*)*σ*¯(*t*), does not hold in general. We are also interested to the question that to what extent the third classical relation in (67) can be generalized when *V* is not smooth. However, it seems that Theorem 3.2 tells us nothing in this context while the following result gives the general

(*t*)) = −*p*(*t*) tell us by the first-order adjoint equation (18).

*c*¯(*t*,*e*)*c*¯(*t*,*e*)�*π*(*de*)

*s*,*y*;*u*¯

) > *V*(*t* � , *x*� ), ∀(*t* � , *x*� �

*π*(*de*), *a*.*e*.*t* ∈ [*s*, *T*], **P***-a*.*s*.,

) �= (*t*, *<sup>x</sup>*) <sup>∈</sup> [*s*, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n.*

(*t*)), *p*(*t*), *P*(*t*)), *a*.*e*.*t* ∈ [*s*, *T*], **P***-a*.*s*.,

(68)

(69)

�

(*t*−)) = −*γ*(*t*, ·), *a*.*e*.*t* ∈ [*s*, *T*], **P***-a*.*s*.

(67)

(*t*)) equal to −*γ*(*t*, ·)

⎧ ⎪⎪⎪⎨ *Vx*(*t*, *x*¯

*Vxx*(*t*, *x*¯

*Vx*(*t*, *x*¯

*s*,*y*;*u*¯

*s*,*y*;*u*¯

*s*,*y*;*u*¯

which is just that *Vx*(*t*, *x*¯*s*,*y*;*u*¯

0 ≤ *tr* �

> + � **E** �

*or, equivalently,*

G(*t*, *x*¯ *s*,*y*;*u*¯

relationship among *p*, *q*, *γ*(·), *P*, *σ*¯ and *c*¯(·).

*σ*¯(*t*)�[*q*(*t*) − *P*(*t*)*σ*¯(*t*)]

(*t*), *u*¯(*t*)) ≥ *G*(*t*, *x*¯

*where <sup>ψ</sup>*<sup>1</sup> <sup>∈</sup> *<sup>C</sup>*1,2([0, *<sup>T</sup>*] <sup>×</sup> **<sup>R</sup>***n*)*, such that <sup>ψ</sup>*1(*<sup>t</sup>*

**Proposition 3.3** *Under the assumption of Theorem 3.4, we have*

� − 1 2 *tr* � *P*(*t*) � **E**

*s*,*y*;*u*¯

*ψ*1(*t*, *x* + *c*¯(*t*,*e*)) − *ψ*1(*t*, *x*) + �*p*(*t*) + *γ*(*t*,*e*), *c*¯(*t*,*e*)�

(*t*), *u*¯(*t*), −*ψ*1(*t*, *x*¯

� , *x*�

⎪⎪⎪⎩

of this chapter.

In this chapter, we have derived the relationship between the maximum principle and dynamic programming principle for the stochastic optimal control problem of jump diffusions. Without involving any derivatives of the value function, relations among the adjoint processes, the generalized Hamiltonian and the value function are derived in the language of viscosity solutions and the associated super-subjets. The conditions under which the above results are valid are very mild and reasonable. The results in this chapter bridge an important gap in the literature.

## **Author details**

Jingtao Shi *Shandong University, P. R. China*

#### **5. References**


[7] Framstad, N.C., Øksendal, B. & Sulem, A. (2004). A sufficient stochastic maximum principle for optimal control of jump diffusions and applications to finance. *Journal of Optimization Theory and Applications*, Vol. 121, No. 1, 77–98. (Errata (2005), this journal, Vol. 124, No. 2, 511–512.)

**Iterations for a General Class of Discrete-Time**

Additional information is available at the end of the chapter

Ivan Ivanov

http://dx.doi.org/10.5772/45718

*<sup>X</sup>*(*i*) = <sup>P</sup>(*i*, **<sup>X</sup>**) :<sup>=</sup> <sup>∑</sup>*<sup>r</sup>*

E*i*(**X**) =

*N* ∑ *j*=1

where *R*(*i*, **X**) = *R*(*i*) + ∑*<sup>r</sup>*

(*X*(1),..., *X*(*N*)) and

**1. Introduction**

**Riccati-Type Equations: A Survey and Comparison**

The discrete-time linear control systems have been applied in the wide area of applications such as engineering, economics, biology. Such type systems have been intensively considered in the control literature in both the deterministic and the stochastic framework. The stability and optimal control of stochastic differential equations with Markovian switching has recently received a lot of attention, see Freiling and Hochhaus [8], Costa, Fragoso, and Marques [2], Dragan and Morozan [4, 5]. The equilibrium in these discrete-time stochastic systems can be found via the maximal solution of the corresponding set of discrete-time Riccati equations. We consider a set of discrete-time generalized Riccati equations that arise in quadratic optimal control of discrete-time stochastic systems subjected to both state-dependent noise and Markovian jumps, i.e. the discrete-time Markovian jump linear systems (MJLS). The iterative method to compute the maximal and stabilizing solution of wide class of discrete-time

We study a problem for computing the maximal symmetric solution to the following set of

*<sup>l</sup>*=<sup>0</sup> *Bl*(*i*)*T*E*i*(**X**)*Al*(*i*) + *<sup>L</sup>*(*i*)*T*), *<sup>i</sup>* <sup>=</sup> 1, . . . , *<sup>N</sup>* ,

*<sup>l</sup>*=<sup>0</sup> *Bl*(*i*)*T*E*i*(**X**)*Bl*(*i*) and <sup>E</sup>(**X**) = (E1(**X**),..., <sup>E</sup>*N*(**X**)) with **<sup>X</sup>** <sup>=</sup>

©2012 Ivanov, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*pij X*(*j*), *X*(*j*) is an *n* × *n* matrix , for *i* = 1, . . . , *N*.

©2012 Ivanov, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

(1)

**Chapter 8**

*<sup>l</sup>*=<sup>0</sup> *Al*(*i*)*T*E*i*(**X**)*Al*(*i*) + *<sup>C</sup>T*(*i*)*C*(*i*)

*<sup>l</sup>*=<sup>0</sup> *Al*(*i*)*T*E*i*(**X**)*Bl*(*i*) + *<sup>L</sup>*(*i*))

nonlinear equations is derived by Dragan, Morozan and Stoica [6, 7].

discrete-time generalized algebraic Riccati equations (DTGAREs):

<sup>×</sup>*R*(*i*, **<sup>X</sup>**)−<sup>1</sup> (∑*<sup>r</sup>*

<sup>−</sup>(∑*<sup>r</sup>*

