Counterexamples to mean square almost periodicity of the solutions of some SDEs with almost periodic coefficients

TL;DR

This paper demonstrates that solutions to certain stochastic differential equations with almost periodic coefficients are not mean square almost periodic, challenging previous claims, although they may still be almost periodic in distribution.

Contribution

It provides a counterexample to the claim that solutions are mean square almost periodic, clarifying the nature of solutions to SDEs with almost periodic coefficients.

Findings

Solutions are not mean square almost periodic

Solutions can be almost periodic in distribution

Challenges previous claims in the literature

Abstract

We show that, contrarily to what is claimed in some papers, the nontrivial solutions of some stochastic differential equations with almost periodic coefficients are never mean square almost periodic (but they can be almost periodic in distribution).