Asymptotic separation between solutions of Caputo fractional stochastic differential equations

TL;DR

This paper proves the global existence and uniqueness of solutions for Caputo fractional stochastic differential equations with order between 0.5 and 1, and shows that solutions diverge asymptotically, implying non-negative Lyapunov exponents.

Contribution

It introduces a new approach using a temporally weighted norm to analyze solutions of Caputo fractional stochastic differential equations, establishing asymptotic separation and Lyapunov properties.

Findings

Solutions are globally unique and exist under Lipschitz conditions.

Asymptotic distance between solutions decays no faster than a specific polynomial rate.

Mean square Lyapunov exponent of solutions is always non-negative.

Abstract

Using a temporally weighted norm we first establish a result on the global existence and uniqueness of solutions for Caputo fractional stochastic differential equations of order $\alpha\in(\frac{1}{2},1)$ whose coefficients satisfy a standard Lipschitz condition. For this class of systems we then show that the asymptotic distance between two distinct solutions is greater than $t^{-\frac{1-\alpha}{2\alpha}-\eps}$ as $t \to \infty$ for any $\eps>0$. As a consequence, the mean square Lyapunov exponent of an arbitrary non-trivial solution of a bounded linear Caputo fractional stochastic differential equation is always non-negative.