Crank-Nicolson scheme for stochastic differential equations driven by fractional Brownian motions

TL;DR

This paper analyzes the convergence rates of the Crank-Nicolson scheme for multidimensional stochastic differential equations driven by fractional Brownian motions, revealing slower rates due to interactions among the driving processes.

Contribution

It establishes new convergence rate results for the Crank-Nicolson scheme in multidimensional fractional SDEs, highlighting the impact of interactions between multiple fractional Brownian motions.

Findings

For m=1 with zero drift, the rate is n^{-2H}.

For m=1 with non-zero drift, the rate is n^{-1/2 - H}.

For m>1, the rate is n^{1/2 - 2H}.

Abstract

We study the Crank-Nicolson scheme for stochastic differential equations (SDEs) driven by multidimensional fractional Brownian motion $(B^{1}, \dots, B^{m})$ with Hurst parameter $H \in (\frac 12,1)$. It is well-known that for ordinary differential equations with proper conditions on the regularity of the coefficients, the Crank-Nicolson scheme achieves a convergence rate of $n^{-2}$, regardless of the dimension. In this paper we show that, due to the interactions between the driving processes $ B^{1}, \dots, B^{m} $, the corresponding Crank-Nicolson scheme for $m$-dimensional SDEs has a slower rate than for the one-dimensional SDEs. Precisely, we shall prove that when $m=1$ and when the drift term is zero, the Crank-Nicolson scheme achieves the exact convergence rate $n^{-2H}$, while in the case $m=1$ and the drift term is non-zero, the exact rate turns out to be $n^{-\frac12 -H}$. In the general case when $m>1$, the exact rate equals $n^{\frac12 -2H}$. In all these cases the limiting distribution of the leading error is proved to satisfy some linear SDE driven by Brownian motions independent of the given fractional Brownian motions.