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

Weiguo Liu and Jiaowan Luo

Communications in Statistics - Theory and Methods, 2017, vol. 46, issue 15, 7427-7443

Abstract: Consider a class of mixed stochastic differential equation (SDE) involving both a Brownian motion and a fractional Brownian motion with Hurst parameter H∈(12,1).$H\in (\frac{1}{2},1).$ We get the mean square rate of convergence δ12$\delta ^{\frac{1}{2}}$ by using a modified Euler method, here δ is the diameter of partition. As we know, the classical Euler method has the rate of convergence δ12∧(2H-1)$\delta ^{\frac{1}{2}\wedge (2H-1)}$ for mixed SDE and δ2H − 1 (resp. δH) 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.

Date: 2017
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DOI: 10.1080/03610926.2016.1152487

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