**3. An existence of an invariant function (with Prob.1) for stochastic dynamical system under strong perturbations**

Consider the diffusion Itô equation in **R**<sup>3</sup> with orthogonal random action with respect to the vector of the solution

$$d\mathbf{v}(t) = -\mu \mathbf{v}(t)dt + \frac{b}{|\mathbf{v}(t)|} [\mathbf{v}(t) \times d\mathbf{w}(t)],\tag{2}$$

where **v**∈ **R**<sup>3</sup> , **w** ∈ **R**<sup>3</sup> , and *wi*ð Þ*t* , *i* ¼ 1, 2, 3 are independent Wiener processes. This equation is a specific form of the Langevin equation.

V. Doobko in [1] showed that the system (2) have an invariant function called a first integral of this system:

$$u(t, \mathbf{v}) = \exp\left\{2\mu t\right\} \left(|\mathbf{v}(\mathbf{0})|^2 - \frac{b^2}{\mu}\right).$$

This, in particular, implies that

$$\lim\_{t \to \infty} \left| \mathbf{v}(t) \right|^2 = \frac{b^2}{\mu},$$

i.e. process ∣**v**ð Þ*t* ∣ is a nonrandom function and the random process **v**ð Þ*t* itself is generated in a sphere of constant radius *<sup>b</sup>*ffiffi *μ* p .

Our method differs for other (see, for example, [11]) preliminary in the fact that we construct a system of differential equation with the given first integral under arbitrary initial conditions. Besides, this algorithm is realized as software and it allows us to choose a set of functions for simulation. Moreover, we can construct both a system of stochastic differential equations and a system of deterministic ones. The goal of this chapter is representation of modern approach to describe of

This chapter is structured as follows. Firstly, we show that the invariant functions

for stochastic systems exist. Then, the generalized Itô – Wentzell formula is represented. It is a differentiated rule for Jump-diffusion function under variables which solves the Jump-diffusion equations system. This rule is basic for the necessary and sufficient conditions for the stochastic first integral (or invariant function with probability 1) for the Jump-diffusion equations system. The next step is the construction of the differential equations system using the given invariant functions. It can be applied for stochastic and nonstochastic cases. The concept of PCP1 (Programmed control with Prob. 1) for stochastic dynamical systems is introduced. Finally, we show an application of the stochastic invariant theory for a transit from deterministic model with invariant to the same stochastic model. Several examples of application of this

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

theory are given and confirmed by results of numerical calculations.

Now we introduce the main concepts which we will use below.

Let *w t*ð Þ, *t*∈ ½ Þ 0, ∞ be a Wiener process or a (standard) Brownian motion, i. e.

A *ν*ð Þ *t*, *A* is called a Poisson random measure or standard Poisson measure (PM) if it is non-negative integer random variable with the Poisson distribution *ν*ð Þ� *t*, *A*

,

*<sup>k</sup>*! exp f g �*t*Πð Þ *<sup>A</sup> :*

**2. Notation and preliminaries**

• it has stationary, independent increments,

*Poi t*ð Þ Πð Þ *A* , and it has the properties of measure:

• it has continuous sample paths,

• *ν*ð Þ *t*, *A* ∈ ∪ f g0 , *ν*ð Þ¼ *t*, ∅ 0,

• **E**½ �¼ *ν*ð Þ *t*, A *t*Πð Þ *A* ,

**62**

• for every *t*>0, *w t*ð Þ has a normal N ð Þ 0, *t* distribution,

• every trajectory of *w t*ð Þ is not differentiated for all *t*≥0.

• *<sup>ν</sup>*ð Þ *<sup>t</sup>*, *<sup>A</sup>* is a random variable for every *<sup>t</sup>*∈½ � 0, *<sup>T</sup>* , *<sup>A</sup>* <sup>∈</sup> *<sup>n</sup>*<sup>0</sup>

• if #*A* is a number of random events from set *A* during *t*, then

**<sup>P</sup>***t*ð Þ¼ #*<sup>A</sup>* <sup>¼</sup> *<sup>k</sup>* ð Þ *<sup>t</sup>*Πð Þ *<sup>A</sup> <sup>k</sup>*

• if *A* ∩ *B* ¼ ∅, then *ν*ð Þ¼ *t*, *A* ∪ *B ν*ð Þþ *t*, *A ν*ð Þ *t*, *B* ,

• *w*ð Þ¼ 0 0,

dynamical systems having a set of invariant functions.

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 formula.

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

*Dk*ð Þ *t*, **x**ð Þ*t d*w*k*þ

X*m k*¼1

> X*m k*¼1

By analogy with the terminology proposed earlier, let us call formula (5) "the

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

It turned out that the first integral exists in the theory of stochastic differential

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

whose coefficients satisfy the conditions of the existence and uniqueness of a

*bik t*, **x**ð ÞÞ *t d*w*k*ð Þ*t* , **x**ð*t*, **x**ð ÞÞ 0 j

*<sup>t</sup>*¼<sup>0</sup> <sup>¼</sup> **<sup>x</sup>**ð Þ <sup>0</sup> , � (6)

*<sup>t</sup>*,*<sup>x</sup>* is called a first integral of the

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

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].

X*n j*¼1

*dt* <sup>þ</sup>X*<sup>n</sup> i*¼1

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

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

*bi*,*k*ð Þ*t*

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

*∂*2 *F t*ð Þ , **x** *∂xi∂x <sup>j</sup>*

> � � � � � **x**¼**x**ð Þ*t*

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

� � � � � **x**¼**x**ð Þ*t* þ

*d*w*k*þ

(5)

*<sup>t</sup>*,*x*,*<sup>γ</sup> :* (4)

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

*Invariants for a Dynamical System with Strong Random Perturbations*

*k*¼1

1 2 X*n i*¼1

#

½ � ð*F*ð*t*, **x**ð Þþ*t g*ð*t*, *γ*ÞÞ � *F*ð*t*, **x**ð ÞÞ *t ν*ð Þþ *dt*, *dγ*

generalized Itô – Wentzell formula for the GSDES with PM" (GIWF).

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

then the stochastic differential exists and

*<sup>∂</sup>xi* **<sup>x</sup>**¼**x**ð Þ*<sup>t</sup>* <sup>þ</sup>

*<sup>∂</sup>xi* **<sup>x</sup>**¼**x**ð Þ*<sup>t</sup>*

*G t*ð Þ , **x**ðÞþ*t g t*ð Þ , *γ* , *γ ν*ð Þ *dt*, *dγ :*

� � � � �

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

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

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

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

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

> þ X*n i*¼1

þ ð *Rγ*

þ ð *Rγ*

2 4

*ai*ð Þ*t*

*bi*,*k*ð Þ*t*

**5. A first integral for GSDES**

controls of dynamical systems.

*dxi*ðÞ¼ *<sup>t</sup> ai*ð*t*, **<sup>x</sup>**ð ÞÞ *<sup>t</sup> dt* <sup>þ</sup>X*<sup>m</sup>*

*k*¼1

solution [12]. A nonrandom function *u t*ð Þ , **<sup>x</sup>** <sup>∈</sup>C1,2

SDE (see [20]).

**65**

system of Itô SDE
