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In this article the problem of existence and uniqueness of solutions of stochastic differential equations with jumps and concentration points are solved. The theoretical results are illustrated by one example.

First, consider the works that are relevant to this subject. Note that number of these works are very small, since the existence of points of condensation is not very often encountered in real processes. However, the relevant equations can significantly enhance the understanding of the dynamics of real processes. In addition, this mathematical model can be a very good comparison for the classical model, the ordinary differential equations, stochastic differential equations, functional differential equations, and impulse equations. On the other hand, equations with concentration points cannot be considered as equations with Poison integral, because for these equations points of condensation do not exist with probability 1.

In paper [

On the other hand, Theorem 3.3 [

In [

A significant contribution to the study of impulse systems of differential equations was made by Ukrainian academician Samoilenko. The monograph [

In [

Problem of existence and uniqueness of solution of impulse systems without the concentration points is considered in [

All above-listed papers do not contain concentration points and cannot be used for describing systems with increase on the short time interval resonance.

The problem of existence and uniqueness of solution of dynamic systems with concentration points for determinate dynamic differential equations is solved in [

The sufficient conditions of existence and uniqueness of the solution of the systems of stochastic differential-difference equations with Markov switching with concentration points are shown in this paper. Thus the paper is actual and timely.

Consider stochastic differential-difference equation

Here

Let measured mapping

Consider the case when the concentration point

In this part of the work consider the problem of existence and uniqueness of solutions of stochastic differential-difference equations with impulse perturbations. Note that the conditions considered in this theorem are not elementary, since they have a quite complicated form.

Let

the condition (

the condition

Consider

According to [

Then, according to the theorem’s condition

Consider linear stochastic differential-difference equation

Here

Let us define the values of the parameter

Define the value of

So,

Let us give the R-realization of the problems (

If

If

As shown in Figure

Usually, in mathematical describing of real processes with short-term perturbations evolution one supposes that perturbations are momentary and mathematical model is dynamic system with discontinuous trajectories. In this case the important class of the systems with impulse impact frequency increasing is lost. This paper is one of the important steps in learning of such systems and development of the quality persistence theory and learning the stabilization problem.

On the other hand, this paper significantly expands the class of equalities, for which we can consider the conditions of resistance, existence of periodic and quasi-periodic solutions, and tasks of the optimal control.

The authors declare that there are no conflicts of interest regarding the publication of this article.