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 ∈ (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.
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BibTeX
@misc{abreu2026consistency,
title = {{Consistency and Convergence of the Backward-Euler Scheme for Stochastic Functional Differential Equations Driven by Fractional Brownian Motion}},
author = {Alexander Abreu and Lisandro Fermin and Ernesto Mordecki and Soledad Torres},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37019},
eprint = {2609.37019},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05768392v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Consistency and Convergence of the Backward-Euler Scheme for Stochastic Functional Differential Equations Driven by Fractional Brownian Motion
AU - Alexander Abreu
AU - Lisandro Fermin
AU - Ernesto Mordecki
AU - Soledad Torres
PY - 2026
DA - 2026/09/29
PB - HAL
DO - 10.48550/arxiv.2609.37019
UR - https://hal.science/hal-05768392v1
AB - 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 ∈ (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.
ER -