Consistency and Convergence of the Backward-Euler Scheme for Stochastic Functional Differential Equations Driven by Fractional Brownian Motion

Abstract

We study the backward-Euler scheme for a class of stochastic functional differential equations (SFDEs) with memory driven by a fractional Brownian motion with Hurst parameter H > 1/2. We first establish the local consistency of the method, with a local truncation error of order H -$ρ$- $β$ + 1, and then, as the main result, prove the uniform global pathwise convergence of the numerical approximation, on a set of probability one, with order H -$ρ$ -$β$, for $β$ $\in$ (1-H, 1/2) and 0 < $ρ$ < H -$β$. A uniform high-probability formulation with deterministic constants follows as a corollary. In particular, the convergence order can be taken arbitrarily close to 2H -1, matching the order obtained for the explicit Euler scheme. The convergence proof relies on an exact integral representation of the scheme at mesh points, an a priori bound for the numerical solution, fractional calculus estimates, and a Gronwall inequality for weakly singular kernels.