**9. Stochastic models with invariant function which are based on deterministic model with invariant one**

In this section we consider a few examples for application of the theory above to modeling actual random processes with invariants [23]. Firstly, we consider an example of construction of a differential equation system with the given invariant. Secondly, we study a general scheme for the PCP1 determination. And finally, we show the possibility of construction of stochastic analogues for classical models described by a differential equations system with an invariant function. The suggested method of stochastization is based on both the concept of the first integral for a stochastic differentialItô equations system (SDE) and the theorem for construction of the SDE system using its first integral.

#### **9.1 Construction of a differential equations system**

It is necessary to construct a differential equations system for **X** ∈ **R**<sup>3</sup> , *t*≥0 such that the equality

$$X(t) - Y\_1^2(t) + Y\_2(t) + e^t = 0 \tag{25}$$

is satisfied with Prob.1. The equality (25) means that the differential equations system has a first integral *s t*<sup>ð</sup> , *X t*ð Þ, *<sup>Y</sup>*1ð Þ*<sup>t</sup>* , *<sup>Y</sup>*2ð Þ*<sup>t</sup>* Þ ¼ *X t*ðÞ� *<sup>Y</sup>*<sup>2</sup> <sup>1</sup>ð Þþ*<sup>t</sup> <sup>Y</sup>*2ðÞþ*<sup>t</sup> <sup>e</sup><sup>t</sup>* with initial condition 0, 1, 0 ð Þ <sup>∗</sup> :

$$\kappa(t, X(t), Y\_1(t), Y\_2(t)) \equiv X(t) - Y\_1^2(t) + Y\_2(t) + \varepsilon^t = \kappa(\mathbf{0}, \varkappa(\mathbf{0}), Y\_1(\mathbf{0}), Y\_2(\mathbf{0})).$$

We have

$$B\_k(\cdot) = q\_{00}(\cdot) \det \begin{bmatrix} \overrightarrow{e}\_1 & \overrightarrow{e}\_2 & \overrightarrow{e}\_3 \\ \mathbf{1} & -2\mathbf{y}\_1 & \mathbf{1} \\ f\_1(\cdot) & f\_2(\cdot) & f\_3(\cdot) \end{bmatrix} = q\_{00}(\cdot) \begin{bmatrix} -2\mathbf{y}\_1 f\_3(\cdot) - f\_2(\cdot) \\ -f\_3(\cdot) + f\_1(\cdot) \\ f\_2(\cdot) + 2\mathbf{y}\_1 f\_1(\cdot) \end{bmatrix} = \begin{bmatrix} b\_{1k}(\cdot) \\ b\_{2k}(\cdot) \\ b\_{3k}(\cdot) \end{bmatrix}.$$

where a column matrix *R t*ð Þ , *x* with components *ri*ð Þ *t*, *x* , *i* ¼ f g 1, … , *n* , is defined

! *<sup>o</sup>* <sup>þ</sup>X*<sup>n</sup> i*¼1

!

<sup>1</sup> … *e*

<sup>2</sup> ⋯ *e*

*ri*ð Þ *t*, *x e* ! *i*,

> ! *n*

… *<sup>∂</sup>s t*ð Þ , *<sup>x</sup> ∂xn*

*<sup>o</sup>* of a matrix *H t*ð Þ , *x* and

*<sup>i</sup>*, related to Poisson measure, is

! *n*

*<sup>∂</sup><sup>x</sup> <sup>j</sup>* , and *φij*ð Þ¼ *t*, *y*ð Þ �, *γ*

*:* (23)

(24)

<sup>⋯</sup> *<sup>∂</sup>s t*ð Þ , *<sup>y</sup>*ð Þ �, *<sup>γ</sup> ∂yn*

*<sup>∂</sup>hi*ð Þ *<sup>t</sup>*, *<sup>x</sup>*

, (22)

ð Þ� *t*, *x* det*H t*ð Þ¼ , *x e*

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

*e* !

*<sup>∂</sup>s t*ð Þ , *<sup>x</sup> ∂t*

is a Jacobi matrix for function *Bk*ð Þ *t*, *x* ,

<sup>1</sup> *e*

This solution satisfies the initial condition: *y t*ð Þj , *<sup>x</sup>*, *<sup>γ</sup> <sup>γ</sup>*¼<sup>0</sup> <sup>¼</sup> *<sup>x</sup>*.

*∂ fi* ð Þ *t*, *x*

*<sup>i</sup>*¼<sup>1</sup>*γi*ð Þ *<sup>t</sup>*, *<sup>x</sup>*, *<sup>γ</sup> <sup>e</sup>*

*<sup>o</sup> e* !

> *<sup>∂</sup>s t*ð Þ , *<sup>x</sup> ∂x*<sup>1</sup>

> > !

defined by the representation Θð Þ¼ *t*, *x*, *γ y t*ð Þ� , *x*, *γ x*, where *y t*ð Þ , *x*, *γ* is a

!

*<sup>∂</sup>s t*ð Þ , *<sup>y</sup>*ð Þ �, *<sup>γ</sup> ∂y*2

The arbitrary functions *fij* ¼ *fij*ð Þ *t*, *x* , *hij* ¼ *hij*ð Þ *t*, *x* , and *φij* ¼ *φij*ð Þ *t*, *y*ð Þ �, *γ* are

*<sup>∂</sup><sup>x</sup> <sup>j</sup>* , *hij*ð Þ¼ *t*, *x*

. Sets of functions *φ<sup>i</sup>* f g ð Þ *t*, *y*ð Þ �, *γ* and the function *g t*ð Þ , *x* together form a

Using this theorem, we can to construct SDE system of different types and ODE system. Choice of arbitrary functions allows us to construct a set of differential equations systems with the given invariant functions. Theorem (7) allows us to introduce a concept of Programmed control with probability 1 for stochastic dynamical system.

**8. Programmed control with Prob. 1 for stochastic dynamical systems**

Definition 1.6 [18, 19]. A PCP1 is called a control of stochastic system which allows the preservation with probability 1 of a constant value for the same function

Let us consider the stochastic nonlinear jump of diffusion equations system: *dX t*ðÞ¼ ð Þ *P t*ð Þþ ,*X t*ð Þ *Z t*ð Þ� , *X t*ð Þ *u t*ð Þ ,*X t*ð Þ *dt* þ ð Þ *B t*ð Þþ ,*X t*ð Þ *K t*ð Þ ,*X t*ð Þ *dw t*ð Þþ

ð Þ *L t*ð Þþ ,*X t*ð Þ, *γ* Λð Þ *t*,*X t*ð Þ *ν*ð Þ *dt*, *dγ* ,

which depends on this systems position for time periods of any length *T*.

*φ*31ð Þ *t*, *y*ð Þ �, *γ φ*32ð Þ *t*, *y*ð Þ �, *γ* ⋯ *φ*3*<sup>n</sup>*ð Þ *t*, *y*ð Þ �, *γ* ⋯ ⋯⋯⋯ *φ<sup>n</sup>*1ð Þ *t*, *y*ð Þ �, *γ φ<sup>n</sup>*2ð Þ *t*, *y*ð Þ �, *γ* ⋯ *φnn*ð Þ *t*, *y*ð Þ �, *γ*

*h*<sup>30</sup> *h*<sup>31</sup> … *h*3*<sup>n</sup>* … ……… *hn*þ1,0 *hn*þ1,1 … *hn*þ1,*<sup>n</sup>*

as follows:

and *<sup>∂</sup>Bk*ð Þ *<sup>t</sup>*, *<sup>x</sup> ∂x* h i

*<sup>∂</sup>y*ð Þ �, *<sup>γ</sup>*

*<sup>∂</sup>φi*ð Þ *<sup>t</sup>*, *<sup>y</sup>*ð Þ �, *<sup>γ</sup> ∂y j*

**70**

*<sup>∂</sup><sup>γ</sup>* <sup>¼</sup> det

defined by the equalities *fij*ð Þ¼ *t*, *x*

þ ð *Rγ*

class of independent functions.

*C*�<sup>1</sup>

det*C t*ð Þ , *x* 6¼ 0, a matrix *H t*ð Þ , *x* is defined as

*H t*ð Þ¼ , *x*

solution of the differential equations system

*e* !

*<sup>∂</sup>s t*ð Þ , *<sup>y</sup>*ð Þ �, *<sup>γ</sup> ∂y*1

**3.** Coefficient <sup>Θ</sup>ð Þ¼ *<sup>t</sup>*,*X*, *<sup>γ</sup>* <sup>P</sup>*<sup>n</sup>*

*C t*ð Þ , *x* is an algebraic adjunct of the element *e*

$$\begin{split} (B\_{k}(\cdot),\nabla\_{x})B\_{k}(\cdot) &= q\_{100}^{2}(\cdot) \left[ \begin{array}{cc} \frac{\partial(-2Y\_{1}f\_{3}(\cdot)-f\_{2}(\cdot))}{\partial x} & \frac{\partial(-2Y\_{1}f\_{3}(\cdot)-f\_{2}(\cdot))}{\partial Y\_{1}} & \frac{\partial(-2Y\_{1}f\_{3}(\cdot)-f\_{2}(\cdot))}{\partial Y\_{2}} \\ \frac{\partial(-f\_{3}(\cdot)+f\_{1}(\cdot))}{\partial x} & \frac{\partial(-f\_{3}(\cdot)+f\_{1}(\cdot))}{\partial Y\_{1}} & \frac{\partial(-f\_{3}(\cdot)+f\_{1}(\cdot))}{\partial Y\_{2}} \\ \frac{\partial(f\_{2}(\cdot)+2Y\_{1}f\_{1}(\cdot))}{\partial x} & \frac{\partial(f\_{2}(\cdot)+2Y\_{1}f\_{1}(\cdot))}{\partial Y\_{1}} & \frac{\partial(f\_{2}(\cdot)+2Y\_{1}f\_{1}(\cdot))}{\partial Y\_{2}} \end{array} \right] \times \\ & \times \begin{bmatrix} -2Y\_{1}(t)f\_{3}(\cdot)-f\_{2}(\cdot) \\ -f\_{3}(\cdot)+f\_{1}(\cdot) \\ f\_{2}(\cdot)+2Y\_{1}(t)f\_{1}(\cdot) \end{bmatrix} = \begin{bmatrix} p\_{1}(\cdot) \\ p\_{2}(\cdot) \\ p\_{3}(\cdot) \end{bmatrix}. \end{split}$$

$$A\_{1}(\cdot) = \frac{\epsilon'(h\_{2}f\_{3} - f\_{2}h\_{3}) + 2Y\_{1}(h\_{0}f\_{3} - f\_{0}h\_{3}) + h\_{0}f\_{2} - f\_{0}h\_{2}}{f\_{2}h\_{3} - h\_{2}f\_{3} + f\_{1}h\_{2} - h\_{3}g\_{2} - 2Y\_{1}(h\_{1}f\_{3} - f\_{1}h\_{3})} + \frac{p\_{1}}{2},$$

$$A\_{2}(\cdot) = \frac{\epsilon'(h\_{3}f\_{1} - f\_{3}h\_{1}) + h\_{0}f\_{3} - f\_{0}h\_{3} + h\_{0}f\_{1} - f\_{0}h\_{1}}{f\_{2}h\_{3} - h\_{2}f\_{3} + f\_{1}h\_{2} - h\_{3}g\_{2} - 2Y\_{1}(h\_{1}f\_{3} - f\_{1}h\_{3})} + \frac{p\_{2}}{2},\tag{26}$$

$$A\_{3}(\cdot) = \frac{\epsilon'(h\_{2}f\_{1} - f\_{2}h\_{1}) - 2Y\_{1}(h\_{0}f\_{1} - f\_{0}h\_{1}) + h\_{0}f\_{2} - f\_{0}h\_{2}}{f\_{2}h\_{3} - h\_{2}f\_{3} + f\_{1}h\_{2} - h\_{3}g\_{2} - 2Y\_{1}(h\_{1}f\_{3} - f\_{1}h\_{3})} + \frac{p\_{3}}{2}.$$

$$\begin{aligned} u(t, Z) &= Z\_1 - Z\_2^2 + Z\_3 + e^t, \\ u(t, Z) - u(t, Z + \mathbf{g}(t, Z, \boldsymbol{\gamma})) &\equiv u(t, Z) - u(t, V) = \mathbf{0}, \\ V &= Z + \mathbf{g}(t, Z, \boldsymbol{\gamma}), \\ \mathbf{g}(t, Z, \boldsymbol{\gamma}) &= V(t, Z, \boldsymbol{\gamma}) - Z, \end{aligned}$$

$$\frac{\partial V(\cdot,\boldsymbol{\chi})}{\partial \boldsymbol{\eta}} = \det \begin{bmatrix} \overrightarrow{\boldsymbol{e}}\_{1} & \overrightarrow{\boldsymbol{e}}\_{2} & \overrightarrow{\boldsymbol{e}}\_{3} \\ \mathbf{1} & -2\mathbf{Z}\_{2} & \mathbf{1} \\ \boldsymbol{\varrho}\_{1}(\cdot,\boldsymbol{\chi}) & \boldsymbol{\varrho}\_{2}(\cdot,\boldsymbol{\chi}) & \boldsymbol{\varrho}\_{3}(\cdot,\boldsymbol{\chi}) \end{bmatrix} = \begin{bmatrix} -2\mathbf{Z}\_{2}\boldsymbol{\varrho}\_{3}(\cdot,\boldsymbol{\chi}) - \boldsymbol{\varrho}\_{2}(\cdot,\boldsymbol{\chi}) \\ \boldsymbol{\varrho}\_{1}(\cdot,\boldsymbol{\chi}) - \boldsymbol{\varrho}\_{3}(\cdot,\boldsymbol{\chi}) \\ \boldsymbol{\varrho}\_{2}(\cdot,\boldsymbol{\chi}) + 2\mathbf{Z}\_{2}\boldsymbol{\varrho}\_{1}(\cdot,\boldsymbol{\chi}), \end{bmatrix},$$

$$\begin{aligned} \mathbf{g}\_1(t, X(t), Y(t, ), \boldsymbol{\chi}) &= -2Y\_1(t)\boldsymbol{\chi} - \boldsymbol{\chi}^2 - \boldsymbol{\chi}(t), \\ \mathbf{g}\_2(t, X(t), Y(t, ), \boldsymbol{\chi}) &= \ln|\boldsymbol{\chi} + \mathbf{1}| - \boldsymbol{\chi} + \mathbf{1} - Y\_1(t), \\ \mathbf{g}\_3(t, X(t), Y(t, ), \boldsymbol{\chi}) &= -\boldsymbol{\chi}^2 + 2Y\_1 \ln|\boldsymbol{\chi} + \mathbf{1}| - Y\_2. \end{aligned}$$

$$\begin{aligned} \label{E} \begin{cases} dX(t) = \frac{\varepsilon'(h\_2 f\_3 - f\_2 h\_3) + 2Y\_1(h\_0 f\_3 - f\_0 h\_3) + h\_0 f\_2 - f\_0 h\_2}{f\_2 h\_3 - h\_2 f\_3 + f\_1 h\_2 - h\_1 g\_2 - 2Y\_1(h\_1 f\_3 - f\_1 h\_3)} \\ \end{cases} \\\\ \begin{aligned} \label{E} \begin{aligned} dY\_1(t) = \frac{\varepsilon'(h\_3 f\_1 - f\_3 h\_1) + h\_0 f\_3 - f\_0 h\_3 + h\_0 f\_1 - f\_0 h\_1}{f\_2 h\_3 - h\_2 f\_3 + f\_1 h\_2 - h\_1 g\_2 - 2Y\_1(h\_1 f\_3 - f\_1 h\_3)} \\ \end{aligned} \\ \begin{aligned} dY\_2(t) = \frac{\varepsilon'(h\_2 f\_1 - f\_2 h\_1) - 2Y\_1(h\_0 f\_1 - f\_0 h\_1) + h\_0 f\_2 - f\_0 h\_2}{f\_2 h\_3 - h\_2 f\_3 + f\_1 h\_2 - h\_2 g\_1 - 2Y\_1(h\_1 f\_3 - f\_1 h\_3)}. \end{aligned} \end{aligned} \end{aligned}$$

Let us consider a classical model

$$\begin{cases} dy\_1(t) = F\_1(t, \mathbf{y}(t))dt, \\ dy\_2(t) = F\_2(t, \mathbf{y}(t))dt, \\ dy\_3(t) = F\_3(t, \mathbf{y}(t))dt, \end{cases} \tag{28}$$

and its restrictsion is

*dS t*ðÞ¼ �*λ*

8

>>>>>>>>>>>>>><

>>>>>>>>>>>>>>:

and

*dx t*ð Þ *dy t*ð Þ *dz t*ð Þ

processes with jumps.

2*e*�*<sup>t</sup>* 0

�2*e*�*<sup>t</sup>*

**Figure 1.**

**75**

*Numerical solution for Eq.(36) without jumps.*

*dI t*ðÞ¼ *λ*

Let the model with strong perturbation be

*<sup>N</sup>* <sup>þ</sup> *<sup>s</sup>*1ð Þ *<sup>t</sup>*, *S t*ð Þ,*I t*ð Þ, *R t*ð Þ � �*dt*<sup>þ</sup>

*<sup>N</sup>* � *<sup>μ</sup>I t*ð Þþ *<sup>s</sup>*2ð Þ *<sup>t</sup>*, *S t*ð Þ,*I t*ð Þ, *R t*ð Þ � �*dt*<sup>þ</sup>

Suppose that the function *u t*ð Þ¼ , *x*, *y*, *z x* þ *y* þ *z* � *N* is a first integral, *v t*ð Þ¼ , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>* <sup>2</sup>*e*�*<sup>t</sup>* <sup>þ</sup> *<sup>x</sup>* and *h t*ð Þ¼ , *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup> <sup>y</sup>* are complementary functions, and *q t*ð Þ¼ , *x*, *y*, *z x* is arbitrary function. The initial condition is: *x*ð Þ¼ 0 1, *y*ð Þ¼ 0 0,

Let us simulate a numerical solution of Eg.(36), where *N* ¼ 1 (for example). **Figure 1** shows simulation for system without jumps, the **Figure 2** shows the

*d*wðÞþ*t*

ð *Rγ*

0 *x t*ð Þ*γ*Þ �*x t*ð Þ*γ*

*ν*ð Þ *dt*, *dγ :* (36)

*z*ð Þ¼ 0 0. Then constructed differential equations system has the form

0 *x t*ð Þ �*x t*ð Þ

<sup>þ</sup>*b*1ð Þ *<sup>t</sup>*, *S t*ð Þ,*I t*ð Þ, *R t*ð Þ *<sup>d</sup>*wðÞþ*<sup>t</sup>* <sup>Ð</sup>

*Invariants for a Dynamical System with Strong Random Perturbations*

<sup>þ</sup>*b*1ð Þ *<sup>t</sup>*, *S t*ð Þ,*I t*ð Þ, *R t*ð Þ *<sup>d</sup>*wðÞþ*<sup>t</sup>* <sup>Ð</sup>

<sup>þ</sup>*b*1ð Þ *<sup>t</sup>*, *S t*ð Þ,*I t*ð Þ, *R t*ð Þ *<sup>d</sup>*wðÞþ*<sup>t</sup>* <sup>Ð</sup>

*dR t*ðÞ¼ ð Þ *μS t*ð Þþ *s*3ð Þ *t*, *S t*ð Þ,*I t*ð Þ, *R t*ð Þ *dt*þ

*S t*ð Þ*I t*ð Þ

*S t*ð Þ*I t*ð Þ

*DOI: http://dx.doi.org/10.5772/intechopen.96235*

*S t*ðÞþ *I t*ðÞþ *R t*ðÞ¼ *N:* (33)

*g*1ð Þ *t*, *S t*ð Þ,*I t*ð Þ, *R t*ð Þ, *γ ν*ð Þ *dt*, *dγ* ,

*g*2ð Þ *t*, *S t*ð Þ,*I t*ð Þ, *R t*ð Þ, *γ ν*ð Þ *dt*, *dγ* ,

*g*3ð Þ *t*, *S t*ð Þ,*I t*ð Þ, *R t*ð Þ, *γ ν*ð Þ *dt*, *dγ* ,

(34)

*Rγ*

*Rγ*

*Rγ*

*u t*ð , *S t*ð Þ,*I t*ð Þ, *R t*ð ÞÞ ¼ *S t*ðÞþ *I t*ðÞþ *R t*ðÞ� *N* � 0*:* (35)

with an invariant *u t*, **y** � �.

Then we construct the GSDE system, taking into account the equality *u t*ð Þ¼ , **x**ð Þ*t u*ð Þ¼ 0, **x**ð Þ 0 *C*:

$$\begin{cases} d\mathbf{x}\_{1}(t) = a\_{1}(t, \mathbf{x}(t))dt + b\_{1}(t, \mathbf{x}(t))d\mathbf{w}(t) + \int \mathbf{g}\_{1}(t, \mathbf{x}(t), \boldsymbol{\eta})\nu(dt, d\boldsymbol{\eta}), \\ d\mathbf{x}\_{2}(t) = a\_{2}(t, \mathbf{x}(t))dt + b\_{2}(t, \mathbf{x}(t))d\mathbf{w}(t) + \int \mathbf{g}\_{2}(t, \mathbf{x}(t), \boldsymbol{\eta})\nu(dt, d\boldsymbol{\eta}), \\ d\mathbf{x}\_{3}(t) = a\_{3}(t, \mathbf{x}(t))dt + b\_{3}(t, \mathbf{x}(t))d\mathbf{w}(t) + \int \mathbf{g}\_{3}(t, \mathbf{x}(t), \boldsymbol{\eta})\nu(dt, d\boldsymbol{\eta}). \end{cases} \tag{29}$$

Hence, the stochastic model has a representation

$$\begin{cases} dy\_1(t) = a\_1(t, \mathbf{y}(t))dt + b\_1(t, \mathbf{y}(t))d\mathbf{w}(t) + \int \mathbf{g}\_1(t, \mathbf{y}(t), \boldsymbol{\eta})\boldsymbol{\nu}(dt, d\boldsymbol{\eta}), \\ dy\_2(t) = a\_2(t, \mathbf{y}(t))dt + b\_2(t, \mathbf{y}(t))d\mathbf{w}(t) + \int \mathbf{g}\_2(t, \mathbf{y}(t), \boldsymbol{\eta})\boldsymbol{\nu}(dt, d\boldsymbol{\eta}), \\ dy\_3(t) = a\_3(t, \mathbf{y}(t))dt + b\_3(t, \mathbf{y}(t))d\mathbf{w}(t) + \int \mathbf{g}\_3(t, \mathbf{y}(t), \boldsymbol{\eta})\boldsymbol{\nu}(dt, d\boldsymbol{\eta}), \\ \mathbf{y}(0) = \mathbf{y}\_0. \end{cases} (30)$$

Further, we determine complementary function which is unit of control functions for PCP1:

$$\begin{cases} \boldsymbol{\varsigma}\_{1}(t, \mathbf{y}(t)) = \boldsymbol{a}\_{1}(t, \mathbf{y}(t)) - \boldsymbol{F}\_{1}(t, \mathbf{y}(t)),\\ \boldsymbol{\varsigma}\_{2}(t, \mathbf{y}(t)) = \boldsymbol{a}\_{2}(t, \mathbf{y}(t)) - \boldsymbol{F}\_{2}(t, \mathbf{y}(t)),\\ \boldsymbol{\varsigma}\_{3}(t, \mathbf{y}(t)) = \boldsymbol{a}\_{3}(t, \mathbf{y}(t)) - \boldsymbol{F}\_{3}(t, \mathbf{y}(t)).\end{cases} \tag{31}$$

Finally, we have constructed stochastic analogue for classical model described by a differential equations system and having an invariant function.
