### Generalization of Asymptotic Behavior of Nonautonomous Stochastic Differential Equation

Олена Анатоліївна Тимошенко

#### Abstract

Background. The study of the asymptotic behavior of solutions of stochastic differential equations is one of the main places in many sections of insurance and financial mathematics, economics, management theory since stochastic differential equations, as an effective model of random process is the basis for the study of random phenomena.

Objective. In this paper we consider the almost sure asymptotic behavior of the solution of the nonautonomous stochastic differential equation.

Methods. We proposed a method to study the y-asymptotic properties of a solution of a stochastic differential equation by comparison with a solution of an ordinary differential equations obtained by dropping the stochastic part. We also use of the theory of pseudo-regularly varying functions.

Results. We investigate the asymptotic behavior of solutions stochastic differential equations and establish sufficient conditions that provide different types of asymptotic behavior of a random process.

Conclusions. Stochastic models approximate the real processes much better than deterministic ones, however, deterministic modelling has been preferred to stochastic one because of much greater ease of computability. The presented result enabled comparing properties of solution a stochastic differential equation with a solution of an ordinary differential equation.

#### Keywords

Stochastic differential equation; Wiener process; Asymptotic behavior

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DOI: https://doi.org/10.20535/1810-0546.2016.4.71649

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