Almost periodic solutions for stochastic differential equations with exponential dichotomy driven by Levy noise

TL;DR

This paper investigates the existence and uniqueness of almost periodic solutions for semilinear stochastic differential equations driven by Lévy noise with exponential dichotomy, demonstrating conditions under which solutions are almost periodic in distribution.

Contribution

It establishes new conditions for the existence, uniqueness, and almost periodicity in distribution of solutions to Lévy-driven stochastic differential equations with exponential dichotomy.

Findings

Existence and uniqueness of bounded solutions under certain conditions.

Conditions for solutions to be almost periodic in distribution.

Illustrative examples demonstrating the theoretical results.

Abstract

In this paper, we study almost periodic solutions for semilinear stochastic differential equations driven by L\'{e}vy noise with exponential dichotomy property. Under suitable conditions on the coefficients, we obtain the existence and uniqueness of bounded solutions. Furthermore, this unique bounded solution is almost periodic in distribution under slightly stronger conditions. We also give two examples to illustrate our results.