**6. Necessary and sufficient conditions for the stochastic first integral**

Lemma 1.1. If *ρ*ð Þ *t*, **x**, *ω* is a stochastic kernel of an integral invariant of *n* th order of a stochastic process **x**ð Þ*t* starting from a point **x**ð Þ 0 then, for every *t*, it satisfies the equality

$$
\rho(t, \mathbf{x}(t, \mathbf{x}(0)), \boldsymbol{\alpha}) \mathcal{J}(t, \mathbf{x}(0), \boldsymbol{\alpha}) = \rho(0, \mathbf{x}(0)),
$$

where J ð Þ *t*, **x**ð Þ 0 , *ω* is the Jacobian of transition from **x**ð Þ*t* to **x**ð Þ 0 .

Definition 1.3 A set of kernels of integral invariants of *n*th order is called complete if any other function that is the kernel of this integral invariant can be presented as a function of the elements of this set.

In [9] it is shown that a system of GSDE (7) whose coefficients satisfy the conditions in (8), has a complete set of kernels consisting of ð Þ *n* þ 1 functions.

Suppose that *ρl*ð Þ *t*, **x**,*ω* 6¼ 0, *l* ¼ 1, … , *m*, *m* ≤ *n* þ 1 are kernels of the integral invariant (9). Lemma 1.1 implies that, for any *<sup>l</sup>* 6¼ *<sup>n</sup>* <sup>þ</sup> 1, the ratio *<sup>ρ</sup>l*ð Þ *<sup>t</sup>*, **<sup>x</sup>**ð Þ *<sup>t</sup>*, **<sup>y</sup>** ,*<sup>ω</sup> <sup>ρ</sup>n*þ<sup>1</sup>ð Þ *<sup>t</sup>*, **<sup>x</sup>**ð Þ *<sup>t</sup>*, **<sup>y</sup>** ,*<sup>ω</sup>* is a constant depending only on the initial condition **x**ð Þ¼ 0 **y** for every solution **x**ð Þ*t* to the GSDE (7) because

$$\frac{\rho\_s(t, \mathbf{x}(t, \mathbf{y}), \boldsymbol{\omega})}{\rho\_{n+1}(t, \mathbf{x}(t, \mathbf{y}), \boldsymbol{\omega})} = \frac{\rho\_s(\mathbf{0}, \mathbf{y})}{\rho\_{n+1}(\mathbf{0}, \mathbf{y})}.\tag{15}$$

Since for some realization *ω*<sup>1</sup> we have

$$u(t, \mathbf{x}(t, \mathbf{x}(\mathbf{0}))) \equiv \frac{\rho\_l(t, \mathbf{x}(t, \mathbf{x}(\mathbf{0})), \boldsymbol{\omega}\_1)}{\rho\_s(t, \mathbf{x}(t, \mathbf{x}(\mathbf{0})), \boldsymbol{\omega}\_1)} = \frac{\rho\_l(\mathbf{0}, \mathbf{x}(\mathbf{0}))}{\rho\_s(\mathbf{0}, \mathbf{x}(\mathbf{0}))} \equiv u(0, \mathbf{x}(\mathbf{0})),$$

Theorem 1.2 Let **x**ð Þ*t* be a solution to the GSDES (7) with conditions (8). A

<sup>2</sup> *<sup>b</sup> j k <sup>t</sup>*, **<sup>x</sup>**<sup>Þ</sup> *<sup>∂</sup>bi k*ð Þ *<sup>t</sup>*, **<sup>x</sup>**

*∂x <sup>j</sup>* � i <sup>¼</sup> <sup>0</sup>

3.*u t*ð Þ� , **x** *u t*ð , **x** þ *g t*ð Þ , **x**, *γ* Þ ¼ 0 for any *γ* ∈*R<sup>γ</sup>* in the entire domain of definition

Theorem (6) allows us to obtain a method for construction of differential

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

*X*ð Þ¼ 0 *x*0, *t*≥ 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

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:

!

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

! *<sup>o</sup>*, *e* ! 1, … , *e* ! *n*

*e* !

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

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

*Rγ*

Θð Þ *t*, *X t*ð Þ, *γ ν*ð Þ *dt*, *dγ*

n o defines an orthogonal

*<sup>i</sup>* ð Þ *k* ¼ f g 1, … , *m* , which determine

! *n*

� *Bk*ð*t*, *x*Þg,

9

>>>>>>>>>>=

, (20)

(21)

>>>>>>>>>>;

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

<sup>1</sup> … *e*

*f* <sup>31</sup> … *f* <sup>3</sup>*<sup>n</sup>* ……… *f <sup>n</sup>*<sup>1</sup> … *f nn*

*<sup>∂</sup>Bk*ð Þ *<sup>t</sup>*, *<sup>x</sup> ∂x* � � (19)

equations systems on the basis of the given set of invariant functions.

*<sup>t</sup>*,*<sup>x</sup>* is a first integral of system (7) if and only if it

,

nonrandom function *u t*ð Þ , **<sup>x</sup>** <sup>∈</sup>C1,2

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

*<sup>∂</sup>xi ai*ð Þ� *<sup>t</sup>*, **<sup>x</sup>** <sup>1</sup>

*<sup>∂</sup>xi* ¼ 0, for all *k* ¼ 1, *m*,

*Invariants for a Dynamical System with Strong Random Perturbations*

the construction of a system of differential equations.

*dX t*ðÞ¼ *A t*ð Þ ,*X t*ð Þ *dt* <sup>þ</sup> *B t*ð Þ , *X t*ð Þ *dw t*ðÞþ <sup>ð</sup>

columns of the matrix *B t*ð Þ , *x* , belong to a set

8

>>>>>>>>>><

>>>>>>>>>>:

*Bk*ð Þ *t*, *x* ∈ *qoo*ð Þ� *t*, *x* det

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

*A t*ð Þ , *x* ∈ *R t*ð Þþ , *x*

(

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

1 2 X*n k*¼1

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

respect to all its variables. Assume the set *e*

**1.** Functions *Bk*ð Þ¼ *<sup>t</sup>*, *<sup>x</sup>* <sup>P</sup>*<sup>n</sup>*

**69**

h

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

satisfies the conditions:

*<sup>∂</sup><sup>t</sup>* <sup>þ</sup> *<sup>∂</sup>u t*ð Þ , **<sup>x</sup>**

of the process.

**invariant functions**

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

2.*bi k*ð Þ *t*, **x**

and it means, that *dtu t*ð Þ¼ , **x**ð Þ*t* 0.

Definition 1.4 A random function *u t*ð Þ , **x**, *ω* defined on the same probability space as a solution to (7) is referred to as a stochastic first integral of the system (7) of Itô Ë† GSDE with NCM if the following condition holds with probability 1:

$$u(t, \mathbf{x}(t, \mathbf{x}(0), \boldsymbol{\omega})) = u(0, \mathbf{x}(0)) \quad \text{almost surely}$$

for every solution **x**ð Þ *t*, **x**ð Þ 0 , *ω* to (7).

For practical purposes, for example, to construct program controls for a dynamical system under strong random perturbations, the presence of a concrete realization is important, i.e., the parameter *ω* is absent in what follows. In this connection, we introduce one more notion.

Definition 1.5 A nonrandom function *u t*ð Þ , **x** is called a first integral of the system of GSDE (7) if it preserves a constant value with probability 1 for every realization of a random process **x**ð Þ*t* that is a solution to this system:

$$
\mu(t, \mathbf{x}(t, \mathbf{x}(0))) = \mu(0, \mathbf{x}(0)) \quad \text{almost surely.}
$$

Thus, a stochastic first integral includes all trajectories (or realizations) of the random process while the first integral is related to one realization.

Construct an equation for *u t*ð Þ , **x**, *ω* using the relation

$$
\ln u\_{\boldsymbol{\epsilon}}(\mathbf{t}, \mathbf{x}, \boldsymbol{\alpha}) = \ln \rho\_{\boldsymbol{\epsilon}}(\mathbf{t}, \mathbf{x}, \boldsymbol{\alpha}) - \ln \rho\_{\boldsymbol{l}}(\mathbf{t}, \mathbf{x}, \boldsymbol{\alpha}), \tag{16}
$$

as a result of assertion (15). Let us differentiate ln *ρ*ð Þ *t*, **x** (omit *ω*) using generalized Itô – Wentzell formula:

$$\begin{split}d\_{t}\ln\rho(t,\mathbf{x}) &= \frac{1}{\rho(t,\mathbf{x})}\ddot{d}\_{t}\rho(t,\mathbf{x}) - \frac{1}{2\rho^{2}(t,\mathbf{x})} \left(-\frac{\partial(\rho(t,\mathbf{x})b\_{ik}(t,\mathbf{x}))}{\partial\mathbf{x}\_{i}}\right)^{2}dt + \\ &+ \int\_{R\_{\gamma}} \left[\ln\left\{\rho\_{\imath}(t,\mathbf{x}-\mathbf{g}(t,\mathbf{x}(t,\mathbf{y}),\gamma),\gamma)\mathcal{J}\left(\mathbf{x}^{-1}(t,\mathbf{x},\gamma)\right)\right\} - \ln\rho\_{\imath}(t,\mathbf{x})\right] \nu(dt,d\gamma), \end{split} \tag{17}$$

where ~ *dtρ*ð Þ *t*, **x** is the right side of Eq.(14) without the integral expression. Having written down the equations for ln *ρs*ð Þ *t*, **x** and ln *ρl*ð Þ *t*, **x** , and taking into account this result and Eq.(16), we obtain:

$$d\_t u(t, \mathbf{x}, \boldsymbol{\omega}) = \left[ -a\_i(t, \mathbf{x}) \frac{\partial u(t, \mathbf{x}, \boldsymbol{\omega})}{\partial \mathbf{x}\_i} + \frac{1}{2} b\_{ik}(t, \mathbf{x}) b\_{jk}(t, \mathbf{x}) \frac{\partial^2 u(t, \mathbf{x}, \boldsymbol{\omega})}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} \right] - $$

$$-b\_{ik}(t, \mathbf{x}) \frac{\partial}{\partial \mathbf{x}\_i} \left( b\_{jk}(t, \mathbf{x}) \frac{\partial u(t, \mathbf{x}, \boldsymbol{\omega})}{\partial \mathbf{x}\_j} \right) \Big] dt - b\_{ik}(t, \mathbf{x}) \frac{\partial u(t, \mathbf{x}, \boldsymbol{\omega})}{\partial \mathbf{x}\_i} d\mathbf{w}\_k(t) + \tag{18}$$

$$+ \int\_{R\_\gamma} \left[ u\left(t, \mathbf{x} - \mathbf{g}\left(t, \mathbf{x}^{-1}(t, \mathbf{x}, \boldsymbol{\gamma}), \boldsymbol{\omega}\right) - u(t, \mathbf{x}, \boldsymbol{\omega}) \right] \nu(dt, d\boldsymbol{\gamma}),$$

which means that a stochastic first integral *u t*ð Þ , **x**, *ω* of the Itô generalized Eq. (7) is a solution to the GSDE (18).

For a first integral which is a nonrandom function of one realization, the differential is also defined by an equation of the form of (18).

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

Theorem 1.2 Let **x**ð Þ*t* be a solution to the GSDES (7) with conditions (8). A nonrandom function *u t*ð Þ , **<sup>x</sup>** <sup>∈</sup>C1,2 *<sup>t</sup>*,*<sup>x</sup>* is a first integral of system (7) if and only if it satisfies the conditions:

$$\begin{aligned} \mathbf{1}. \frac{\partial \boldsymbol{u}(t, \mathbf{x})}{\partial t} + \frac{\partial \boldsymbol{u}(t, \mathbf{x})}{\partial \boldsymbol{x}\_i} \left[ \boldsymbol{a}\_i(t, \mathbf{x}) - \frac{1}{2} \boldsymbol{b}\_{\, jk} \left( t, \mathbf{x} \right) \frac{\partial \boldsymbol{b}\_{ik}(t, \mathbf{x})}{\partial \boldsymbol{x}\_j} \right] &= \mathbf{0}, \\\\ \mathbf{2}. \boldsymbol{b}\_{ik}(t, \mathbf{x}) \frac{\partial \boldsymbol{u}(t, \mathbf{x})}{\partial \boldsymbol{x}\_i} &= \mathbf{0}, \text{ for all } k = \overline{\mathbf{1}, m}, \end{aligned}$$

Since for some realization *ω*<sup>1</sup> we have

and it means, that *dtu t*ð Þ¼ , **x**ð Þ*t* 0.

for every solution **x**ð Þ *t*, **x**ð Þ 0 , *ω* to (7).

we introduce one more notion.

alized Itô – Wentzell formula:

1 *<sup>ρ</sup>*ð Þ *<sup>t</sup>*, **<sup>x</sup>** <sup>~</sup>

account this result and Eq.(16), we obtain:

*b j k*ð Þ *t*, **x**

*u t*, **<sup>x</sup>** � *g t*, **<sup>x</sup>**�<sup>1</sup>

*dtu t*ð Þ¼ � , **x**,*ω ai*ð Þ *t*, *x*

*∂ ∂xi* �

Eq. (7) is a solution to the GSDE (18).

ln *<sup>ρ</sup><sup>s</sup> <sup>t</sup>*, **<sup>x</sup>** � *g t*, **<sup>x</sup>** *<sup>t</sup>*, **<sup>y</sup>** � �, *<sup>γ</sup>*Þ, *<sup>γ</sup>* � �<sup>J</sup> **<sup>x</sup>**�<sup>1</sup>

*dt* ln *ρ*ð Þ¼ *t*, **x**

þ ð *Rγ*

**68**

where ~

�*bi k*ð Þ *t*, **x**

þ ð *Rγ*

*u t*ð Þ� , **<sup>x</sup>**ð Þ *<sup>t</sup>*, **<sup>x</sup>**ð Þ <sup>0</sup> *<sup>ρ</sup>l*ð Þ *<sup>t</sup>*, **<sup>x</sup>**ð Þ *<sup>t</sup>*, **<sup>x</sup>**ð Þ <sup>0</sup> , *<sup>ω</sup>*<sup>1</sup>

of a random process **x**ð Þ*t* that is a solution to this system:

random process while the first integral is related to one realization.

Construct an equation for *u t*ð Þ , **x**, *ω* using the relation

*dtρ*ð Þ� *t*, **x**

*<sup>∂</sup>u t*ð Þ , *<sup>x</sup>*,*<sup>ω</sup> ∂xi*

*<sup>∂</sup>u t*ð Þ , **<sup>x</sup>**,*<sup>ω</sup> ∂x <sup>j</sup>*

� ��

differential is also defined by an equation of the form of (18).

*ρs*ð Þ *t*, **x**ð Þ *t*, **x**ð Þ 0 , *ω*<sup>1</sup>

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

Definition 1.4 A random function *u t*ð Þ , **x**, *ω* defined on the same probability space as a solution to (7) is referred to as a stochastic first integral of the system (7) of Itô Ë† GSDE with NCM if the following condition holds with probability 1:

*u t*ð , **x**ð Þ *t*, **x**ð Þ 0 , *ω* Þ ¼ *u*ð Þ 0, **x**ð Þ 0 almost surely

For practical purposes, for example, to construct program controls for a dynamical system under strong random perturbations, the presence of a concrete realization is important, i.e., the parameter *ω* is absent in what follows. In this connection,

Definition 1.5 A nonrandom function *u t*ð Þ , **x** is called a first integral of the system of GSDE (7) if it preserves a constant value with probability 1 for every realization

*u t*ð Þ¼ , **x**ð Þ *t*, **x**ð Þ 0 *u*ð Þ 0, **x**ð Þ 0 almost surely*:*

Thus, a stochastic first integral includes all trajectories (or realizations) of the

as a result of assertion (15). Let us differentiate ln *ρ*ð Þ *t*, **x** (omit *ω*) using gener-

1

ð Þ *<sup>t</sup>*, **<sup>x</sup>**, *<sup>γ</sup>* � � � � � ln *<sup>ρ</sup>s*ð Þ *<sup>t</sup>*, **<sup>x</sup>** � �*ν*ð Þ *dt*, *<sup>d</sup><sup>γ</sup>* , �

Having written down the equations for ln *ρs*ð Þ *t*, **x** and ln *ρl*ð Þ *t*, **x** , and taking into

þ 1 2

ð Þ *<sup>t</sup>*, **<sup>x</sup>**, *<sup>γ</sup>* , *<sup>γ</sup>*Þ,*<sup>ω</sup>* � � � *u t*ð Þ , **<sup>x</sup>**,*<sup>ω</sup>* � �*ν*ð Þ *dt*, *<sup>d</sup><sup>γ</sup>* , �

which means that a stochastic first integral *u t*ð Þ , **x**, *ω* of the Itô generalized

For a first integral which is a nonrandom function of one realization, the

*dtρ*ð Þ *t*, **x** is the right side of Eq.(14) without the integral expression.

ln *us*ð Þ¼ *t*, **x**,*ω* ln *ρs*ð Þ� *t*, **x**,*ω* ln *ρl*ð Þ *t*, **x**,*ω* , (16)

<sup>2</sup>*ρ*<sup>2</sup>ð Þ *<sup>t</sup>*, **<sup>x</sup>** � *<sup>∂</sup>*ð Þ *<sup>ρ</sup>*ð Þ *<sup>t</sup>*, **<sup>x</sup>** *bi k*ð Þ *<sup>t</sup>*, **<sup>x</sup>**

*bi k*ð*t*, *x*Þ*b j k*ð*t*, *x*Þ

*dt* � *bi k*ð Þ *t*, **x**

*∂xi* � �<sup>2</sup>

*∂*2

*<sup>∂</sup>u t*ð Þ , **<sup>x</sup>**, *<sup>ω</sup> ∂xi*

*u t*ð Þ , *x*,*ω ∂xi∂x <sup>j</sup>*

�

*d*w*k*ð Þþ*t*

*dt*þ

(17)

(18)

<sup>¼</sup> *<sup>ρ</sup>l*ð Þ 0, **<sup>x</sup>**ð Þ <sup>0</sup>

*<sup>ρ</sup>s*ð Þ 0, **<sup>x</sup>**ð Þ <sup>0</sup> � *<sup>u</sup>*ð Þ 0, **<sup>x</sup>**ð Þ <sup>0</sup> ,

3.*u t*ð Þ� , **x** *u t*ð , **x** þ *g t*ð Þ , **x**, *γ* Þ ¼ 0 for any *γ* ∈*R<sup>γ</sup>* in the entire domain of definition of the process.

Theorem (6) allows us to obtain a method for construction of differential equations systems on the basis of the given set of invariant functions.
