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

2013/06/06 by Yaozhong Hu, Hu, Yaozhong, Yanghui Liu +3 · 1 citation
Economics, Econometrics and Finance · Social Sciences · #FOS: Mathematics #Financial Risk and Volatility Modeling #Insurance, Mortality, Demography, Risk Management #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1306.1458

openalex publication_date 2013/06/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

For a stochastic differential equation driven by a fractional Brownian motion with Hurst parameter H> \frac12 it is known that the classical Euler scheme has the rate of convergence 2H-1. In this paper we introduce a new numerical scheme 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 \frac 12 (the rate of Euler scheme for classical Brownian motion). The rate of weak convergence is also deduced for this scheme. The main tools are fractional calculus and Malliavin calculus. We also apply our approach to the classical Euler scheme.

Citations

• An inequality of the Hölder type, connected with Stieltjes integration

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