Square-mean almost automorphic solutions for some stochastic differential equations

TL;DR

This paper introduces the concept of square-mean almost automorphy for stochastic processes and proves existence, uniqueness, and stability of such solutions for certain stochastic differential equations.

Contribution

It is the first to define square-mean almost automorphy and establish solution properties for stochastic differential equations under this new framework.

Findings

Existence and uniqueness of solutions under specified conditions

Asymptotic stability of the solutions in square-mean sense

Introduction of the square-mean almost automorphy concept

Abstract

The concept of square-mean almost automorphy for stochastic processes is introduced. The existence and uniqueness of square-mean almost automorphic solutions to some linear and non-linear stochastic differential equations are established provided the coefficients satisfy some conditions. The asymptotic stability of the unique square-mean almost automorphic solution in square-mean sense is discussed.