Mittag-Leffler Euler integrator for a stochastic fractional order equation with additive noise

TL;DR

This paper introduces the Mittag-Leffler Euler integrator, a novel numerical method for stochastic fractional order equations with additive noise, demonstrating improved convergence rates over traditional methods through theoretical analysis and numerical validation.

Contribution

The paper proposes a new Mittag-Leffler Euler integrator for stochastic fractional equations, achieving higher convergence rates than existing methods like backward Euler with convolution quadrature.

Findings

Temporal convergence rate nearly doubled compared to backward Euler method.

Numerical experiments confirm theoretical convergence rates.

Spectral Galerkin method effectively discretizes spatial variables.

Abstract

Motivated by fractional derivative models in viscoelasticity, a class of semilinear stochastic Volterra integro-differential equations, and their deterministic counterparts, are considered. A generalized exponential Euler method, named here as the Mittag-Leffler Euler integrator, is used for the temporal discretization, while the spatial discretization is performed by the spectral Galerkin method. The temporal rate of strong convergence is found to be (almost) twice compared to when the backward Euler method is used together with a convolution quadrature for time discretization. Numerical experiments that validate the theory are presented.