Error Distribution for One-Dimensional Stochastic Differential Equation Driven By Fractional Brownian Motion

TL;DR

This paper investigates the asymptotic errors of numerical schemes for one-dimensional stochastic differential equations driven by fractional Brownian motion, providing new evaluations and extending error analysis to cases with low Hurst exponents.

Contribution

It introduces a new evaluation method for convergence and asymptotic errors, fully determines the Milstein method's error, and extends error analysis of the Crank-Nicolson scheme to lower Hurst exponents.

Findings

Complete asymptotic error characterization for Milstein method.

New error analysis for Crank-Nicolson scheme with 1/4<H≤1/3.

Improved convergence conditions for numerical schemes.

Abstract

This paper deals with asymptotic errors, limit theorems for errors between numerical and exact solutions of stochastic differential equation (SDE) driven by one-dimensional fractional Brownian motion (fBm). The Euler-Maruyama, higher-order Milstein, and Crank-Nicolson schemes are among the most studied numerical schemes for SDE (fSDE) driven by fBm. Most previous studies of asymptotic errors have derived specific asymptotic errors for these schemes as main theorems or their corollary. Even in the one-dimensional case, the asymptotic error was not determined for the Milstein or the Crank-Nicolson method when the Hurst exponent is less than or equal to $1/3$ with a drift term. We obtained a new evaluation method for convergence and asymptotic errors. This evaluation method improves the conditions under which we can prove convergence of the numerical scheme and obtain the asymptotic error under the same conditions. We have completely determined the asymptotic error of the Milstein method for arbitrary orders. In addition, we have newly determined the asymptotic error of the Crank-Nicolson method for $1/4<H\leq 1/3$.