On asymptotically periodic solution of a stochastic differential equation

TL;DR

This paper investigates the existence of asymptotically periodic solutions in stochastic differential equations driven by Brownian motion, introducing the concept of {}-periodic limit processes and providing criteria for such solutions.

Contribution

It introduces the concept of {}-periodic limit processes and establishes criteria for asymptotically {}-periodic mild solutions in stochastic differential equations.

Findings

Established criteria for asymptotically {}-periodic solutions

Introduced the concept of {}-periodic limit processes

Provided an example demonstrating the theoretical results

Abstract

In this paper, we first introduce the concept and properties of {\omega}- periodic limit process. Then we apply specific criteria obtained to investigate asymptotically {\omega}-periodic mild solutions of a Stochastic Differential Equation driven by a Brownian motion. Finally, we give an example to show usefulness of the theoritical results that we obtain in the paper.