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

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 $H\in (\frac{1}{2},1)$. 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 $n^{-\frac{1}{2}-H}$. In the general case when $m>1$, the exact rate equals $n^{\frac{1}{2}-2H}$. 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.

38 pages, 2 figures

References in corpus (3)

• Central limit theorems for sequences of multiple stochastic integrals
• The rate of convergence of Euler approximations for solutions of stochastic differential equations driven by fractional Brownian motion
• Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions

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• Stochastic modified equations for symplectic methods applied to rough Hamiltonian systems based on the Wong--Zakai approximation
• Optimal convergence rate of modified Milstein scheme for SDEs with rough fractional diffusions