Modified Euler approximation scheme for stochastic differential equations driven by fractional Brownian motions

摘要

For a stochastic differential equation driven by a fractional Brownian motionwith Hurst parameter $H> \frac12$ it is known that the classical Euler schemehas the rate of convergence $2H-1$. In this paper we introduce a new numericalscheme which is closer to the classical Euler scheme for diffusion processes,in the sense that it has the rate of convergence $2H-\frac12$. In particular,the rate of convergence becomes $\frac 12$ when $H$ is formally set to $\frac12$ (the rate of Euler scheme for classical Brownian motion). The rate of weakconvergence is also deduced for this scheme. The main tools are fractionalcalculus and Malliavin calculus. We also apply our approach to the classicalEuler scheme.