**4. The generalized Itô – Wentzell formula for jump-diffusion function**

The rules for constructing stochastic differentials, e.g., the change rule, are very important in the theory of stochastic random processes. These are Itô's formula [13, 14] for the differential of a nonrandom function of a random process and the Itô – Wentzell<sup>2</sup> formula [15] enabling us to construct the differential of a function which per se is a solution to a stochastic equation. Many articles address the derivation of these formulas for various classes of processes by extending Itô's formula and the Itô – Wentzell formula to a larger class of functions.

The next level is to obtain a new formula for the generalized Itô Equation [14] which involves Wiener and Poisson components. In 2002, V. Doobko presented [7] a generalization of stochastic differentials of random functions satisfying GSDES with CPM based on expressions for the kernels of integral invariants (only the ideas of a possible proof) were sketched in [7]. The result is called" the generalized Itô – Wentzell formula".

In contrast to [7], the generalized Itô – Wentzell formula for the noncentered Poisson measure was represented in [9, 16, 17]. The proof [9] of the generalized Itô – Wentzell formula uses the method of stochastic integral invariants and equations for their kernels. In this case the requirement on the character of the Poisson distribution is only a general restriction, as the knowledge of its explicit form is unnecessary. Other proofs in [16, 17] are based on traditional stochastic analysis and the use of approximations to random functions related to stochastic differential equations by averaging their values at each point.

The generalized Itô – Wentzell formula relying on the kernels of integral invariants [9] requires stricter conditions on the coefficients of all equations under consideration: the existence of second derivatives. The reason is that the kernels of invariants for differential equations exist under certain restrictions on the coefficients.

Since the random function *F t*ð Þ , **x**ð Þ*t* has representation as stochastic diffusion Itô equation with jumps, we can use the generalized Itô – Wentzell formula, proved by us by several methods in accordance with different conditions for the equations coefficients. Now we consider only one case.

We will use the following notation: <sup>C</sup>*<sup>s</sup> <sup>y</sup>* is the space of functions having continuous derivatives of order *<sup>s</sup>* with respect to *<sup>y</sup>*, <sup>C</sup>*<sup>s</sup>* <sup>0</sup>ð Þ*y* is the space of bounded functions having bounded continuous derivatives of order *s* with respect to *y*.

Theorem 1.1 **(generalized Itô – Wentzell formula)**. Consider the real function *F t*ð Þ , **<sup>x</sup>** <sup>∈</sup>C1,2 *<sup>t</sup>*,*<sup>x</sup>* , ð Þ *<sup>t</sup>*, **<sup>x</sup>** <sup>∈</sup> ½ �� 0, *<sup>T</sup> <sup>n</sup>* with generalized stochastic differential of the form

$$d\_t F(t, \mathbf{x}) = Q(t, \mathbf{x})dt + \sum\_{k=1}^{m} D\_k(t, \mathbf{x})d\mathbf{w}\_k(t) + \int\_{R\_\gamma} G(t, \mathbf{x}, \gamma)\nu(dt, d\gamma) \tag{3}$$

whose coefficients satisfy the conditions:

$$Q(t, \mathbf{x}) \in \mathcal{C}^{1,2}\_{t, \mathbf{x}}, \quad D\_k(t, \mathbf{x}) \in \mathcal{C}^{1,2}\_{t, \mathbf{x}}, \quad G(t, \mathbf{x}, \boldsymbol{\gamma}) \in \mathcal{C}^{1,2,1}\_{t, \mathbf{x}, \boldsymbol{\gamma}}.$$

<sup>2</sup> Different variants of transliteration of this formula name: Itô – Wentcell, Itô – Venttcel', Itô – Ventzell

*Invariants for a Dynamical System with Strong Random Perturbations DOI: http://dx.doi.org/10.5772/intechopen.96235*

If a random process **x**ð Þ*t* obeys (1) and its coefficients satisfy the conditions

$$a\_i(t, \mathbf{x}) \in \mathcal{C}^{1,1}\_{t, \mathbf{x}}, \quad b\_{\vec{\eta}}(t, \mathbf{x}) \in \mathcal{C}^{1,2}\_{t, \mathbf{x}}, \quad \mathbf{g}\_i(t, \mathbf{x}, \boldsymbol{\gamma}) \in \mathcal{C}^{1,2,1}\_{t, \mathbf{x}, \boldsymbol{\gamma}}.\tag{4}$$

then the stochastic differential exists and

In [4, 5, 10] it is shown that invariant function exists for other stochastic equations of Langevin type. To obtain this result, it is necessary to use the Itô's

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

**4. The generalized Itô – Wentzell formula for jump-diffusion function**

and the Itô – Wentzell formula to a larger class of functions.

equations by averaging their values at each point.

coefficients. Now we consider only one case. We will use the following notation: <sup>C</sup>*<sup>s</sup>*

ous derivatives of order *<sup>s</sup>* with respect to *<sup>y</sup>*, <sup>C</sup>*<sup>s</sup>*

*dtF t*ð Þ¼ , **<sup>x</sup>** *Q t*ð Þ , **<sup>x</sup>** *dt* <sup>þ</sup>X*<sup>m</sup>*

whose coefficients satisfy the conditions:

*Q t*ð Þ , **<sup>x</sup>** <sup>∈</sup>C1,2

having bounded continuous derivatives of order *s* with respect to *y*.

*k*¼1

*<sup>t</sup>*,*<sup>x</sup>* , *Dk*ð Þ *<sup>t</sup>*, **<sup>x</sup>** <sup>∈</sup> <sup>C</sup>1,2

The rules for constructing stochastic differentials, e.g., the change rule, are very important in the theory of stochastic random processes. These are Itô's formula [13, 14] for the differential of a nonrandom function of a random process and the Itô – Wentzell<sup>2</sup> formula [15] enabling us to construct the differential of a function which per se is a solution to a stochastic equation. Many articles address the derivation of these formulas for various classes of processes by extending Itô's formula

The next level is to obtain a new formula for the generalized Itô Equation [14] which involves Wiener and Poisson components. In 2002, V. Doobko presented [7] a generalization of stochastic differentials of random functions satisfying GSDES with CPM based on expressions for the kernels of integral invariants (only the ideas of a possible proof) were sketched in [7]. The result is called" the generalized Itô –

In contrast to [7], the generalized Itô – Wentzell formula for the noncentered Poisson measure was represented in [9, 16, 17]. The proof [9] of the generalized Itô – Wentzell formula uses the method of stochastic integral invariants and equations for their kernels. In this case the requirement on the character of the Poisson distribution is only a general restriction, as the knowledge of its explicit form is unnecessary. Other proofs in [16, 17] are based on traditional stochastic analysis and the use of approximations to random functions related to stochastic differential

The generalized Itô – Wentzell formula relying on the kernels of integral invariants [9] requires stricter conditions on the coefficients of all equations under consideration: the existence of second derivatives. The reason is that the kernels of invariants for differential equations exist under certain restrictions on the

Since the random function *F t*ð Þ , **x**ð Þ*t* has representation as stochastic diffusion Itô equation with jumps, we can use the generalized Itô – Wentzell formula, proved by us by several methods in accordance with different conditions for the equations

Theorem 1.1 **(generalized Itô – Wentzell formula)**. Consider the real function

*Dk*ð Þ *t*, **x** *d*w*k*ð Þþ*t*

<sup>2</sup> Different variants of transliteration of this formula name: Itô – Wentcell, Itô – Venttcel', Itô – Ventzell

*<sup>t</sup>*,*<sup>x</sup>* , ð Þ *<sup>t</sup>*, **<sup>x</sup>** <sup>∈</sup> ½ �� 0, *<sup>T</sup> <sup>n</sup>* with generalized stochastic differential of the form

ð *Rγ*

*<sup>t</sup>*,*<sup>x</sup>* , *G t*ð Þ , **<sup>x</sup>**, *<sup>γ</sup>* <sup>∈</sup>C1,2,1

*<sup>y</sup>* is the space of functions having continu-

<sup>0</sup>ð Þ*y* is the space of bounded functions

*G t*ð Þ , **x**, *γ ν*ð Þ *dt*, *dγ* (3)

*<sup>t</sup>*,*x*,*<sup>γ</sup> :*

formula.

Wentzell formula".

coefficients.

*F t*ð Þ , **<sup>x</sup>** <sup>∈</sup>C1,2

**64**

$$\begin{split} d\_{l}F(t,\mathbf{x}(t)) &= \mathbf{Q}(t,\mathbf{x}(t))dt + \sum\_{k=1}^{m} D\_{k}(t,\mathbf{x}(t))d\mathbf{w}\_{k} + \\ &+ \left[\sum\_{i=1}^{n} a\_{i}(t)\frac{\partial F(t,\mathbf{x})}{\partial \mathbf{x}\_{i}}\bigg|\_{\mathbf{x}=\mathbf{x}(t)} + \frac{1}{2}\sum\_{i=1}^{n}\sum\_{j=1}^{n}\sum\_{k=1}^{m}b\_{i,k}(t)b\_{j,k}(t)\frac{\partial^{2}F(t,\mathbf{x})}{\partial \mathbf{x}\_{i}\partial \mathbf{x}\_{j}}\bigg|\_{\mathbf{x}=\mathbf{x}(t)} + \\ &+ \sum\_{i=1}^{n}b\_{i,k}(t)\frac{\partial D\_{k}(t,\mathbf{x})}{\partial \mathbf{x}\_{i}}\bigg|\_{\mathbf{x}=\mathbf{x}(t)}\bigg|dt + \sum\_{i=1}^{n}\sum\_{k=1}^{m}b\_{i,k}(t)\frac{\partial F(t,\mathbf{x})}{\partial \mathbf{x}\_{i}}\bigg|\_{\mathbf{x}=\mathbf{x}(t)}d\mathbf{w}\_{k} + \\ &+ \int\_{R\_{\gamma}}[(F(t,\mathbf{x}(t)+\mathbf{g}(t,\boldsymbol{\gamma}))-F(t,\mathbf{x}(t)))\nu(dt,d\boldsymbol{\gamma}) + \\ &+ \int\_{R\_{\gamma}}G(t,\mathbf{x}(t)+\mathbf{g}(t,\boldsymbol{\gamma}),\boldsymbol{\gamma})\nu(dt,d\boldsymbol{\gamma}). \end{split} \tag{5}$$

By analogy with the terminology proposed earlier, let us call formula (5) "the generalized Itô – Wentzell formula for the GSDES with PM" (GIWF).

By analogy with the classical Itô and Itô – Wentzell formulas, the generalized Itô – Wentzell formula is promising for various applications. In particular, it helped to obtain equations for the first and stochastic first integrals of the stochastic Itô system [9], equations for the density of stochastic dynamical invariants, Kolmogorov equations for the density of transition probabilities of random processes described by the generalized stochastic Itô differential Equation [8], as well as the construction of program controls with probability 1 for stochastic systems [18, 19].
