Asymptotic error distribution of Mittag--Leffler Euler method for a fractional stochastic differential equation

TL;DR

This paper analyzes the asymptotic distribution of errors in the Mittag--Leffler Euler method when applied to multidimensional fractional stochastic differential equations reformulated as stochastic Volterra equations with matrix-valued kernels.

Contribution

It introduces a novel analysis of the error distribution for numerical methods with non-diagonal matrix-valued kernels, using an auxiliary scheme and stable convergence theory.

Findings

Established the asymptotic error distribution for the MLE method.

First work to analyze numerical methods with non-diagonal matrix-valued kernels.

Bridged the gap between discretized and exact solutions using an auxiliary scheme.

Abstract

In this paper, we investigate the asymptotic distribution of the normalized error for the Mittag--Leffler Euler (MLE) method applied to a class of multidimensional fractional stochastic differential equations. These equations are reformulated as stochastic Volterra equations (SVEs) featuring a non-diagonal, matrix-valued kernel $K(u)=u^{\alpha-1}E_{\alpha,\alpha}(Au^{\alpha})$ with singular exponent $\alpha \in (\frac{1}{2}, 1)$. To enhance computational efficiency, the singular kernel is discretized using the left-rectangle rule, posing technical challenges for the theoretical analysis. To address this, we introduce an auxiliary $K$-undiscretized scheme to bridge the gap between the exact solution and the MLE method, integrating Jacod's stable convergence theory for conditional Gaussian martingales with methodologies developed for SVEs. To the best of our knowledge, this is the first work to establish the asymptotic error distribution for numerical methods incorporating non-diagonal matrix-valued kernels.