Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion

TL;DR

This paper investigates the optimal approximation rates for stochastic differential equations driven by fractional Brownian motion with Hurst parameter greater than 1/2, identifying when perfect approximation is possible and providing an implementable scheme for the best achievable rate.

Contribution

It establishes the optimal convergence rate for approximating solutions and introduces a practical scheme to attain this rate when perfect approximation isn't possible.

Findings

Optimal rate of convergence is either perfect or n^{-H-1/2}.

An implementable scheme achieves the optimal rate in the non-perfect case.

The rate depends on the Hurst parameter H.

Abstract

We study the approximation of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H>1/2$. For the mean-square error at a single point we derive the optimal rate of convergence that can be achieved by any approximation method using an equidistant discretization of the driving fractional Brownian motion. We find that there are mainly two cases: either the solution can be approximated perfectly or the best possible rate of convergence is $n^{-H-1/2},$ where $n$ denotes the number of evaluations of the fractional Brownian motion. In addition, we present an implementable approximation scheme that obtains the optimal rate of convergence in the latter case.