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 a multidimensional fractional Brownian motion with Hurst parameter $\mathit{H}>1/2$. 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 ${\mathit{n}}^{-2}$, regardless of the dimension. In this paper we show that, due to the interactions between the driving processes, the corresponding Crank–Nicolson scheme for m-dimensional SDEs has a slower rate than for one-dimensional SDEs. Precisely, we shall prove that when the fBm is one-dimensional and when the drift term is zero, the Crank–Nicolson scheme achieves the convergence rate ${\mathit{n}}^{-2\mathit{H}}$, and when the drift term is nonzero, the exact rate turns out to be ${\mathit{n}}^{-\frac{1}{2}-\mathit{H}}$. In the general multidimensional case the exact rate equals ${\mathit{n}}^{\frac{1}{2}-2\mathit{H}}$. In all these cases the asymptotic error is proved to satisfy some linear SDE. We also consider the degenerated cases when the asymptotic error equals zero.