Caputo fractional stochastic differential equations: Lipschitz continuity in the fractional order

TL;DR

This paper investigates how solutions to Caputo fractional stochastic differential equations depend continuously on the fractional order, providing explicit convergence rates and demonstrating their optimality.

Contribution

It offers the first explicit estimates for the weak convergence rate of solutions as the fractional order varies, establishing the optimality of these rates.

Findings

Explicit convergence rate estimates for solutions as fractional order varies

Demonstration of the optimality of the convergence rates

Analysis of the continuous dependence of solutions on fractional order

Abstract

In this paper, we consider a class of the Caputo fractional stochastic differential equations of fractional order $\alpha \in (\frac{1}{2},1]$. Our aim is to analyze of the continuous dependence of solutions on the fractional order $\alpha.$ We first provide explicit estimates for the rate of weak convergence the solutions. We then describe the exact asymptotic behavior of this convergence to show that the rate is optimal.