Optimal convergence rate of modified Milstein scheme for SDEs with rough fractional diffusions

TL;DR

This paper establishes the optimal almost sure convergence rate of the modified Milstein scheme for SDEs driven by fractional Brownian motion with Hurst parameter between 1/4 and 1/2, confirming conjectures and revealing new convergence cases.

Contribution

It introduces a novel framework combining rough path theory and stochastic backward error analysis to analyze the convergence of numerical schemes for fractional SDEs.

Findings

Proves the convergence rate $(2H-\frac12)^-$ for the modified Milstein scheme.

Confirms the conjecture for $H\in(\frac13,\frac12)$.

Shows convergence for schemes based on second-order Taylor expansion when $H\in(\frac14,\frac13]$.

Abstract

We combine the rough path theory and stochastic backward error analysis to develop a new framework for error analysis on numerical schemes. Based on our approach, we prove that the almost sure convergence rate of the modified Milstein scheme for stochastic differential equations driven by multiplicative multidimensional fractional Brownian motion with Hurst parameter $H\in(\frac14,\frac12)$ is $(2H-\frac12)^-$ for sufficiently smooth coefficients, which is optimal in the sense that it is consistent with the result of the corresponding implementable approximation of the L\'evy area of fractional Brownian motion. Our result gives a positive answer to the conjecture proposed in [11] for the case $H\in(\frac13,\frac12)$, and reveals for the first time that numerical schemes constructed by a second-order Taylor expansion do converge for the case $H\in(\frac14,\frac13]$.