Modified Euler approximation of stochastic differential equation driven by Brownian motion and fractional Brownian motion

摘要

Consider a class of mixed stochastic differential equation (SDE) involving both a Brownian motion and a fractional Brownian motion with Hurst parameter H is an element of (1/2, 1). We get the mean square rate of convergence delta(1/2) by using a modified Euler method, here delta is the diameter of partition. As we know, the classical Euler method has the rate of convergence delta(1/2 boolean AND(2H-1)) for mixed SDE and delta(2H-1) (resp. delta(H)) for pathwise (resp. Skorokhod) SDE driven only by fBm, which were proved by Mishura and Shevchenko Mishura and Shevchenko (2011) and Mishura and Shevchenko (2008), respectively. Therefore, we obtain a better result than those of them.