Asymptotic error distribution of Mittag--Leffler Euler method for a fractional stochastic differential equation
This work addresses a theoretical gap in numerical analysis for fractional stochastic differential equations, which is incremental but specific to computational mathematics.
The paper tackles the problem of analyzing the asymptotic error distribution for the Mittag--Leffler Euler method applied to fractional stochastic differential equations, reformulated with non-diagonal matrix-valued kernels, and establishes this distribution as a first in the field.
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 $K(u)=u^{α-1}E_{α,α}(Au^α)$ with singular exponent $α\in (\frac{1}{2}, 1)$. 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.