**7. Construction of the differential equations system using the given invariant functions**

The concept of a first integral for a system of stochastic differential equations plays a key role in our theory. In this section, we will use a set of first integrals for the construction of a system of differential equations.

Let us write Eq. (7) in matrix form:

$$dX(t) = A(t, X(t))dt + B(t, X(t))dw(t) + \int\_{R\_{\gamma}} \Theta(t, X(t), \gamma)\nu(dt, d\gamma) \tag{19}$$

$$X(0) = \mathbf{x}\_0, \qquad t \ge \mathbf{0}.$$

Theorem 1.3 [22]. Let *X t*ð Þ be a solution of the Eq. (19) and let a nonrandom function *s t*ð Þ , *x* be continuous together with its first-order partial derivatives with respect to all its variables. Assume the set *e* ! *<sup>o</sup>*, *e* ! 1, … , *e* ! *n* n o defines an orthogonal basis in **<sup>R</sup>**<sup>þ</sup> � **<sup>R</sup>***<sup>n</sup>*. If function *s t*ð Þ , *<sup>x</sup>* is a first integral for the system (19), then the coefficients of Eq. (19) and the function *s t*ð Þ , *x* together are related by the conditions:

**1.** Functions *Bk*ð Þ¼ *<sup>t</sup>*, *<sup>x</sup>* <sup>P</sup>*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*bik*ð Þ *<sup>t</sup>*, *<sup>x</sup> <sup>e</sup>* ! *<sup>i</sup>* ð Þ *k* ¼ f g 1, … , *m* , which determine columns of the matrix *B t*ð Þ , *x* , belong to a set

$$B\_k(t, \mathbf{x}) \in \left\{ q\_{oo}(t, \mathbf{x}) \cdot \det \begin{bmatrix} \overrightarrow{e}\_1 & \dots & \overrightarrow{e}\_n \\ \frac{\partial s(t, \mathbf{x})}{\partial \mathbf{x}\_1} & \dots & \frac{\partial s(t, \mathbf{x})}{\partial \mathbf{x}\_n} \\\\ f\_{31} & \dots & f\_{3n} \\ \end{bmatrix} \right\}, \tag{20}$$

where *qoo*ð Þ *t*, *x* is an arbitrary nonvanishing function,

**2.** Coefficient *A t*ð Þ , *x* belongs to a set of functions defined by

$$A(t, \boldsymbol{x}) \in \left\{ R(t, \boldsymbol{x}) + \frac{1}{2} \sum\_{k=1}^{n} \left[ \frac{\partial B\_{k}(t, \boldsymbol{x})}{\partial \boldsymbol{x}} \right] \cdot B\_{k}(t, \boldsymbol{x}) \right\},\tag{21}$$

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

$$C^{-1}(t, \boldsymbol{x}) \cdot \det H(t, \boldsymbol{x}) = \overrightarrow{\boldsymbol{e}}\_o + \sum\_{i=1}^n r\_i(t, \boldsymbol{x}) \overrightarrow{\boldsymbol{e}}\_i,$$

*C t*ð Þ , *x* is an algebraic adjunct of the element *e* ! *<sup>o</sup>* of a matrix *H t*ð Þ , *x* and det*C t*ð Þ , *x* 6¼ 0, a matrix *H t*ð Þ , *x* is defined as

$$H(t, \mathbf{x}) = \begin{bmatrix} \overrightarrow{\mathbf{e}}\_o & \overrightarrow{\mathbf{e}}\_1 & \dots & \overrightarrow{\mathbf{e}}\_n \\ \frac{\partial \mathbf{s}(t, \mathbf{x})}{\partial t} & \frac{\partial \mathbf{s}(t, \mathbf{x})}{\partial \mathbf{x}\_1} & \dots & \frac{\partial \mathbf{s}(t, \mathbf{x})}{\partial \mathbf{x}\_n} \\ h\_{30} & h\_{31} & \dots & h\_{3n} \\ \dots & \dots & \dots & \dots \\ h\_{n+1, 0} & h\_{n+1, 1} & \dots & h\_{n+1, n} \end{bmatrix},\tag{22}$$

where *P*ð Þ� , *Z*ð Þ� are given matrix functions and *B*ð Þ� , *L*ð Þ� are the functions that may either be known or not. For such systems we construct a unit of programmed control f g *u t*ð Þ , *X t*ð Þ , *K t*ð Þ , *X t*ð Þ , *M t*ð Þ , *X t*ð Þ which allows the system (24) to be on the given manifold f*u t*ð ,*X t*ð ÞÞg ¼ f g *u*ð Þ 0, *x*<sup>0</sup> with Prob. 1 (PCP1) for each *t* ∈½ � 0, *T* ,

Suppose that the nonrandom function *s t*ð Þ , *X t*ð Þ is the first integral for the same stochastic dynamical system. The PCP1 f g *u t*ð Þ ,*X t*ð Þ ,*K t*ð Þ , *X t*ð Þ , *M t*ð Þ , *X t*ð Þ is the

Theorem 1.4 Let a controlled dynamical system be subjected to Brownian perturbations and Poisson jumps. The unit of PCP1 f g *u t*ð Þ , *X t*ð Þ ,*K t*ð Þ , *X t*ð Þ , *M t*ð Þ , *X t*ð Þ , allowing this system to remain with probability 1 on the dynamically structured integral mfd *s t*ð Þ¼ ,*X t*ð Þ , *xo* ,*ω s*ð Þ 0, *xo* , is a solution of the linear equations system (with respect to functions **u**ð Þ *t*, **x**ð Þ*t* ), *K t*ð Þ ,*X t*ð Þ , *M t*ð Þ , *X t*ð Þ which consists of Eq. (19) and Eq. (24). The coefficients of the Eq. (19), (and the coefficients of the Eq. (24) respectively) are determined by the theorem 7. The response to the

We show how the stochastic invariants theory can be applied to solve different

In this section we consider a few examples for application of the theory above to

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

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

It is necessary to construct a differential equations system for **X** ∈ **R**<sup>3</sup>

<sup>1</sup>ðÞþ*t Y*2ðÞþ*t e*

is satisfied with Prob.1. The equality (25) means that the differential equations

<sup>1</sup>ðÞþ*t Y*2ð Þþ*t e*

2 6 4

3 7 <sup>5</sup> <sup>¼</sup> *<sup>q</sup>*00ð Þ�

*X t*ðÞ� *<sup>Y</sup>*<sup>2</sup>

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>

1 �2*y*<sup>1</sup> 1 *f* <sup>1</sup>ð Þ� *f* <sup>2</sup>ð Þ� *f* <sup>3</sup>ð Þ� , *t*≥0 such

*b*1*<sup>k</sup>*ð Þ� *b*2*<sup>k</sup>*ð Þ� *b*3*<sup>k</sup>*ð Þ� 3 7 5*:*

2 6 4

*<sup>t</sup>* <sup>¼</sup> <sup>0</sup> (25)

*<sup>t</sup>* <sup>¼</sup> *<sup>s</sup>*ð Þ 0, *<sup>x</sup>*ð Þ <sup>0</sup> , *<sup>Y</sup>*1ð Þ <sup>0</sup> , *<sup>Y</sup>*2ð Þ <sup>0</sup> *:*

�2*y*<sup>1</sup> *f* <sup>3</sup>ðÞ�� *f* <sup>2</sup>ð Þ� �*f* <sup>3</sup>ðÞþ� *f* <sup>1</sup>ð Þ� *f* <sup>2</sup>ðÞþ� 2*y*<sup>1</sup> *f* <sup>1</sup>ð Þ�

<sup>1</sup>ð Þþ*<sup>t</sup> <sup>Y</sup>*2ðÞþ*<sup>t</sup> <sup>e</sup><sup>t</sup>* with

solution for the algebraic system of linear equations.

*Invariants for a Dynamical System with Strong Random Perturbations*

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

**deterministic model with invariant one**

construction of the SDE system using its first integral.

**9.1 Construction of a differential equations system**

random action is defined completely.

*T* ≤ ∞.

tasks.

that the equality

We have

*Bk*ðÞ¼ � *q*00ð Þ� det

**71**

initial condition 0, 1, 0 ð Þ <sup>∗</sup> :

*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 � *X t*ðÞ� *<sup>Y</sup>*<sup>2</sup>

*e* ! <sup>1</sup> *e* ! <sup>2</sup> *e* ! 3

2 6 4

and *<sup>∂</sup>Bk*ð Þ *<sup>t</sup>*, *<sup>x</sup> ∂x* h i is a Jacobi matrix for function *Bk*ð Þ *<sup>t</sup>*, *<sup>x</sup>* ,

**3.** Coefficient <sup>Θ</sup>ð Þ¼ *<sup>t</sup>*,*X*, *<sup>γ</sup>* <sup>P</sup>*<sup>n</sup> <sup>i</sup>*¼<sup>1</sup>*γi*ð Þ *<sup>t</sup>*, *<sup>x</sup>*, *<sup>γ</sup> <sup>e</sup>* ! *<sup>i</sup>*, related to Poisson measure, is 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 solution of the differential equations system

$$\frac{\partial \boldsymbol{y}(\cdot,\boldsymbol{\chi})}{\partial \boldsymbol{\chi}} = \det \begin{bmatrix} \overrightarrow{\boldsymbol{e}}\_{1} & \overrightarrow{\boldsymbol{e}}\_{2} & \cdots & \overrightarrow{\boldsymbol{e}}\_{n} \\ \frac{\partial \boldsymbol{s}(t,\boldsymbol{\chi}(\cdot,\boldsymbol{\chi}))}{\partial \boldsymbol{\eta}\_{1}} & \frac{\partial \boldsymbol{s}(t,\boldsymbol{\chi}(\cdot,\boldsymbol{\chi}))}{\partial \boldsymbol{\eta}\_{2}} & \cdots & \frac{\partial \boldsymbol{s}(t,\boldsymbol{\chi}(\cdot,\boldsymbol{\chi}))}{\partial \boldsymbol{\eta}\_{n}} \\ \boldsymbol{\varrho}\_{31}(t,\boldsymbol{\chi}(\cdot,\boldsymbol{\chi})) & \boldsymbol{\varrho}\_{32}(t,\boldsymbol{\chi}(\cdot,\boldsymbol{\chi})) & \cdots & \boldsymbol{\varrho}\_{3n}(t,\boldsymbol{\chi}(\cdot,\boldsymbol{\chi})) \\ \cdots & \cdots & \cdots & \cdots \\ \boldsymbol{\varrho}\_{n1}(t,\boldsymbol{\chi}(\cdot,\boldsymbol{\chi})) & \boldsymbol{\varrho}\_{n2}(t,\boldsymbol{\chi}(\cdot,\boldsymbol{\chi})) & \cdots & \boldsymbol{\varrho}\_{nn}(t,\boldsymbol{\chi}(\cdot,\boldsymbol{\chi})) \end{bmatrix}. \tag{23}$$

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>*.

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 defined by the equalities *fij*ð Þ¼ *t*, *x ∂ fi* ð Þ *t*, *x <sup>∂</sup><sup>x</sup> <sup>j</sup>* , *hij*ð Þ¼ *t*, *x <sup>∂</sup>hi*ð Þ *<sup>t</sup>*, *<sup>x</sup> <sup>∂</sup><sup>x</sup> <sup>j</sup>* , and *φij*ð Þ¼ *t*, *y*ð Þ �, *γ <sup>∂</sup>φi*ð Þ *<sup>t</sup>*, *<sup>y</sup>*ð Þ �, *<sup>γ</sup> ∂y j* . 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 class of independent functions.

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.
