**5. A first integral for GSDES**

In the theory of ODE, there are constructed equations to find deterministic functions, first integrals which preserve a constant value with any solutions to the equation. The concept of a first integral plays an important role in theoretical mechanics, for example, to solve inverse problems of mechanics or in constructing controls of dynamical systems.

It turned out that the first integral exists in the theory of stochastic differential equations (SDE) as well. However, there appears an additional classification connected with different interpretations. This gives a first integral for a system of SDE (see [1]), a first direct integral, and a first inverse integral for a system of Itô SDE (see [20]).

Definition 1.1 [1, 3]. Let **x**ð Þ*t* be an *n*-dimensional random process satisfying a system of Itô SDE

$$d\mathbf{x}\_i(t) = a\_i(t, \mathbf{x}(t))dt + \sum\_{k=1}^{m} b\_{ik}\left(t, \mathbf{x}(t)\right) d\mathbf{w}\_k(t), \qquad \mathbf{x}(t, \mathbf{x}(0))|\_{t=0} = \mathbf{x}(0), \tag{6}$$

whose coefficients satisfy the conditions of the existence and uniqueness of a solution [12]. A nonrandom function *u t*ð Þ , **<sup>x</sup>** <sup>∈</sup>C1,2 *<sup>t</sup>*,*<sup>x</sup>* is called a first integral of the

system of SDE if it takes a constant value depending only on **x**ð Þ 0 on any trajectory solution to (6) with probability 1:

Definition 1.2 [3]. A nonnegative random function *ρ*ð Þ *t*, **x**, *ω* is referred to as a stochastic kernel or the stochastic density of a stochastic integral invariant (of *n*th

Note that a substantial difference which made it possible to consider the invariance of the random volume on the basis of a kernel of an integral operator in [3, 7], is that (9) contains a functional factor. Thus, the notion of a kernel of an integral invariant [3] for a system of ordinary differential equations can be regarded as a particular case by taking *f t*ð Þ¼ , **x** 1 and excluding from (7) the randomness

Using the GIWF (5), we obtain equation for the stochastic kernel function [9].

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

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

*<sup>ρ</sup>*ð Þj *<sup>t</sup>*, **<sup>x</sup>**,*<sup>ω</sup> <sup>t</sup>*¼<sup>0</sup> <sup>¼</sup> *<sup>ρ</sup>*ð Þ¼ 0, **<sup>x</sup>**,*<sup>ω</sup> <sup>ρ</sup>*ð Þ 0, **<sup>x</sup>** <sup>∈</sup> <sup>C</sup><sup>2</sup>

∣**x**∣!∞

This result plays a major role in obtaining of equation for the stochastic first

**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

*ρ*ð*t*, **x**ð Þ *t*, **x**ð Þ 0 , *ω*ÞJ ð Þ¼ *t*, **x**ð Þ 0 ,*ω ρ*ð Þ 0, **x**ð Þ 0 ,

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>*

constant depending only on the initial condition **x**ð Þ¼ 0 **y** for every solution **x**ð Þ*t* to

*<sup>ρ</sup><sup>n</sup>*þ<sup>1</sup> *<sup>t</sup>*, **<sup>x</sup>** *<sup>t</sup>*, **<sup>y</sup>** � �,*<sup>ω</sup>* � � <sup>¼</sup> *<sup>ρ</sup><sup>s</sup>* 0, **<sup>y</sup>** � �

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

*ρ<sup>s</sup> t*, **x** *t*, **y** � �, *ω* � �

*ρ*ð Þ¼ 0, **x** 0, lim

*dwk*ð Þ þ ð� *t*

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

<sup>0</sup>ð Þ **x** ,

¼ lim ∣**x**∣!∞

Þ*dt*þ

*<sup>∂</sup>ρ*ð Þ 0, **<sup>x</sup>**, *<sup>ω</sup> ∂xi*

þ

*<sup>∂</sup>ρ*ð Þ 0, **<sup>x</sup>** *∂xi*

¼ 0*:*

*<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

*<sup>ρ</sup><sup>n</sup>*þ<sup>1</sup> 0, **<sup>y</sup>** � � *:* (15)

(14)

order) if conditions (9), (10), and (11) are held.

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

*Invariants for a Dynamical System with Strong Random Perturbations*

determined by the Wiener and Poisson processes.

*∂xi*

ð Þ *<sup>t</sup>*, **<sup>x</sup>**, *<sup>γ</sup>* , *<sup>γ</sup>*, *<sup>ω</sup>*<sup>Þ</sup> � � � J **<sup>x</sup>**�<sup>1</sup>

*dtρ*ð Þ¼� *<sup>t</sup>*, **<sup>x</sup>**,*<sup>ω</sup> <sup>∂</sup>ρ*ð Þ *<sup>t</sup>*, **<sup>x</sup>**,*<sup>ω</sup> bi k*ð Þ *<sup>t</sup>*, **<sup>x</sup>**

þ 1 2

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

*ρ*ð Þ¼ 0, **x**, *ω* lim

∣**x**∣!∞

presented as a function of the elements of this set.

under restrictions

lim ∣*x*∣!∞

integral.

the equality

the GSDE (7) because

**67**

þ ð *Rγ*

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

or, in other words, its stochastic differential is equal to zero: *dtu t*ð Þ¼ , **x**ð Þ*t* 0*:* Another important notion in the theory of deterministic dynamical systems is given by the notion of an integral invariant introduced by Poincaré [21].

As it turned out, there also exist integral invariants for stochastic dynamical systems [2, 3]. In [7] V. Doobko give the concept of a kernel (=density) of a stochastic integral invariant and, based on it, formulate the notion of a stochastic first integral and a first integral as a deterministic function for GSDES with the centered Poisson measure, which makes it possible to compose a list of first integrals for stochastic differential equations.

Consider a random process **<sup>x</sup>**ð Þ*<sup>t</sup>* , **<sup>x</sup>**<sup>∈</sup> *<sup>n</sup>*, which is a solution to GSDES

$$\begin{aligned} d\mathbf{x}\_i(t) &= a\_i(t, \mathbf{x}(t))dt + b\_{ik}(t, \mathbf{x}(t))d\mathbf{w}\_k(t) + \int\_{R\_\gamma} \mathbf{g}\_i(t, \mathbf{x}(t), \boldsymbol{\eta})\nu(dt, d\boldsymbol{\eta}), \\ \mathbf{x}(t) &= \mathbf{x}(t, \mathbf{x}(0), \boldsymbol{\alpha})|\_{t=0} = \mathbf{x}(0), \qquad i = 1, \ldots, n, \qquad t \ge 0, \end{aligned} \tag{7}$$

whose coefficients (in general, random functions) satisfy the conditions of the existence and uniqueness of a solution [12] and the following smoothness conditions:

$$a\_i(t, \mathbf{x}) \in \mathcal{C}^{1,1}\_{t,\mathbf{x}}, \quad b\_{\vec{\eta}}(t, \mathbf{x}) \in \mathcal{C}^{1,2}\_{t,\mathbf{x}}, \quad \mathbf{g}\_i(t, \mathbf{x}, \boldsymbol{\gamma}) \in \mathcal{C}^{1,2,1}\_{t,\mathbf{x},\boldsymbol{\gamma}}.\tag{8}$$

Suppose that *ρ*ð Þ *t*, **x**,*ω* is a random function connected with any deterministic function *f t*ð Þ , **<sup>x</sup>** <sup>∈</sup> <sup>S</sup> <sup>⊂</sup>C1,2 <sup>0</sup> ð Þ *t*, **x** by the relations

$$\int\_{\mathbb{R}^{n}} \rho(t, \mathbf{x}, \omega) f(t, \mathbf{x}) d\hat{\Gamma}(\mathbf{x}) = \int\_{\mathbb{R}^{n}} \rho(\mathbf{0}, \mathbf{y}) f(t, \mathbf{x}(t, \mathbf{y})) d\hat{\Gamma}(\mathbf{y}) \tag{9}$$

$$\int\_{\mathbb{R}^n} \rho(\mathbf{0}, \mathbf{x}) d\hat{\Gamma}(\mathbf{x}) = \mathbf{1},\tag{10}$$

$$\lim\_{|\mathbf{x}| \to \infty} \rho(\mathbf{0}, \mathbf{x}, \boldsymbol{\omega}) = \lim\_{|\mathbf{x}| \to \infty} \rho(\mathbf{0}, \mathbf{x}) = \mathbf{0}, \qquad d\hat{\Gamma}(\mathbf{x}) = \prod\_{i=1}^{n} d\mathbf{x}\_i,\tag{11}$$

where **<sup>y</sup>**≔**x**ð Þ <sup>0</sup> , and **<sup>x</sup>** *<sup>t</sup>*, **<sup>y</sup>** � � is a solution to (7), and *<sup>ω</sup>* is a random event. In the particular case when *f t*ð Þ¼ , **x** 1, conditions (9) and (10) imply that

$$\int\_{\mathbb{R}^{\mathbf{x}}} \rho(t, \mathbf{x}, a) d\hat{\Gamma}(\mathbf{x}) = \int\_{\mathbb{R}^{\mathbf{x}}} \rho(\mathbf{0}, \mathbf{y}) d\hat{\Gamma}(\mathbf{y}) = 1,\tag{12}$$

i.e., for the random function *ρ*ð Þ *t*, **x**,*ω* , there exists a nonrandom functional preserving a constant value:

$$\int\_{\mathbb{R}^n} \rho(t, \mathbf{x}, a) d\hat{\Gamma}(\mathbf{x}) = \mathbf{1}.\tag{13}$$

Then, with conditions (10) and (11), Eq. (9) can be regarded as a stochastic integral invariant, and the function (t, x) can be viewed as its density.

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

Definition 1.2 [3]. A nonnegative random function *ρ*ð Þ *t*, **x**, *ω* is referred to as a stochastic kernel or the stochastic density of a stochastic integral invariant (of *n*th order) if conditions (9), (10), and (11) are held.

Note that a substantial difference which made it possible to consider the invariance of the random volume on the basis of a kernel of an integral operator in [3, 7], is that (9) contains a functional factor. Thus, the notion of a kernel of an integral invariant [3] for a system of ordinary differential equations can be regarded as a particular case by taking *f t*ð Þ¼ , **x** 1 and excluding from (7) the randomness determined by the Wiener and Poisson processes.

Using the GIWF (5), we obtain equation for the stochastic kernel function [9].

$$d\_t\rho(t,\mathbf{x},\boldsymbol{\omega}) = -\frac{\partial\rho(t,\mathbf{x},\boldsymbol{\omega})b\_{ik}(t,\mathbf{x})}{\partial\mathbf{x}\_i}dw\_k(t) + (-\frac{\partial(\rho(t,\mathbf{x},\boldsymbol{\omega})a\_i(t,\mathbf{x}))}{\partial\mathbf{x}\_i} + 1)$$

$$+ \frac{1}{2}\frac{\partial^2(\rho(t,\mathbf{x},\boldsymbol{\omega})b\_{ik}(t,\mathbf{x})b\_{jk}(t,\mathbf{x}))}{\partial\mathbf{x}\_i\partial\mathbf{x}\_j})dt + \tag{14}$$

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

under restrictions

system of SDE if it takes a constant value depending only on **x**ð Þ 0 on any trajectory

*u t*, **<sup>x</sup>**ð Þ¼ *<sup>t</sup>*, **<sup>x</sup>**ð Þ <sup>0</sup> *<sup>u</sup>*ð Þ 0, **<sup>x</sup>**ð Þ <sup>0</sup> almost surely, �

or, in other words, its stochastic differential is equal to zero: *dtu t*ð Þ¼ , **x**ð Þ*t* 0*:* Another important notion in the theory of deterministic dynamical systems is

As it turned out, there also exist integral invariants for stochastic dynamical systems [2, 3]. In [7] V. Doobko give the concept of a kernel (=density) of a stochastic integral invariant and, based on it, formulate the notion of a stochastic first integral and a first integral as a deterministic function for GSDES with the centered Poisson measure, which makes it possible to compose a list of first

given by the notion of an integral invariant introduced by Poincaré [21].

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

Consider a random process **<sup>x</sup>**ð Þ*<sup>t</sup>* , **<sup>x</sup>**<sup>∈</sup> *<sup>n</sup>*, which is a solution to GSDES

**<sup>x</sup>**ðÞ¼ *<sup>t</sup> x t*ð j , **<sup>x</sup>**ð Þ <sup>0</sup> , *<sup>ω</sup>*<sup>Þ</sup> *<sup>t</sup>*¼<sup>0</sup> <sup>¼</sup> **<sup>x</sup>**ð Þ <sup>0</sup> , *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>n</sup>*, *<sup>t</sup>*<sup>≥</sup> 0,

existence and uniqueness of a solution [12] and the following smoothness

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

<sup>0</sup> ð Þ *t*, **x** by the relations

*<sup>ρ</sup>*ð Þ *<sup>t</sup>*, **<sup>x</sup>**,*<sup>ω</sup> f t*ð Þ , **<sup>x</sup>** *<sup>d</sup>*Γ^ð Þ¼ *<sup>x</sup>*

*ρ*ð Þ¼ 0, **x**,*ω* lim

ð *n*

∣**x**∣!∞

*<sup>ρ</sup>*ð Þ *<sup>t</sup>*, **<sup>x</sup>**, *<sup>ω</sup> <sup>d</sup>*Γ^ð Þ¼ **<sup>x</sup>**

ð *n*

integral invariant, and the function (t, x) can be viewed as its density.

where **<sup>y</sup>**≔**x**ð Þ <sup>0</sup> , and **<sup>x</sup>** *<sup>t</sup>*, **<sup>y</sup>** � � is a solution to (7), and *<sup>ω</sup>* is a random event. In the particular case when *f t*ð Þ¼ , **x** 1, conditions (9) and (10) imply that

> ð *n*

i.e., for the random function *ρ*ð Þ *t*, **x**,*ω* , there exists a nonrandom functional

Then, with conditions (10) and (11), Eq. (9) can be regarded as a stochastic

whose coefficients (in general, random functions) satisfy the conditions of the

Suppose that *ρ*ð Þ *t*, **x**,*ω* is a random function connected with any deterministic

ð *n* ð *Rγ gi*

*<sup>t</sup>*,*<sup>x</sup>* , *gi*

ð Þ *t*, **x**ð Þ*t* , *γ ν*ð Þ *dt*, *dγ* ,

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

*ρ* 0, **y** � �*f t*, **x** *t*, **y** � � � � *d*Γ^ **y**

*<sup>ρ</sup>*ð Þ¼ 0, **<sup>x</sup>** 0, *<sup>d</sup>*Γ^ð Þ¼ **<sup>x</sup>** <sup>Y</sup>*<sup>n</sup>*

*ρ* 0, **y** � �*d*Γ^ **y**

*<sup>ρ</sup>*ð Þ 0, **<sup>x</sup>** *<sup>d</sup>*Γ^ð Þ¼ **<sup>x</sup>** 1, (10)

*<sup>ρ</sup>*ð Þ *<sup>t</sup>*, **<sup>x</sup>**,*<sup>ω</sup> <sup>d</sup>*Γ^ð Þ¼ **<sup>x</sup>** <sup>1</sup>*:* (13)

*i*¼1

� � <sup>¼</sup> 1, (12)

(7)

*<sup>t</sup>*,*x*,*<sup>γ</sup> :* (8)

� � (9)

*dxi*, (11)

solution to (6) with probability 1:

integrals for stochastic differential equations.

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

conditions:

function *f t*ð Þ , **<sup>x</sup>** <sup>∈</sup> <sup>S</sup> <sup>⊂</sup>C1,2

ð *n*

lim ∣**x**∣!∞

preserving a constant value:

**66**

ð *n*

*dxi*ðÞ¼ *t ai*ð Þ *t*, **x**ð Þ*t dt* þ *bik*ð Þ *t*, **x**ð Þ*t d*w*k*ðÞþ*t*

$$\rho(t, \mathbf{x}, \boldsymbol{\alpha})|\_{t=0} = \rho(\mathbf{0}, \mathbf{x}, \boldsymbol{\alpha}) = \rho(\mathbf{0}, \mathbf{x}) \in \mathcal{C}\_0^2(\mathbf{x}),$$

$$\lim\_{|\mathbf{x}| \to \infty} \rho(\mathbf{0}, \mathbf{x}, \boldsymbol{\alpha}) = \lim\_{|\mathbf{x}| \to \infty} \rho(\mathbf{0}, \mathbf{x}) = \mathbf{0}, \quad \lim\_{|\mathbf{x}| \to \infty} \frac{\partial \rho(\mathbf{0}, \mathbf{x}, \boldsymbol{\alpha})}{\partial \mathbf{x}\_i} = \lim\_{|\mathbf{x}| \to \infty} \frac{\partial \rho(\mathbf{0}, \mathbf{x})}{\partial \mathbf{x}\_i} = \mathbf{0}.$$

This result plays a major role in obtaining of equation for the stochastic first integral.
