2013/06/06 by Yaozhong Hu, Hu, Yaozhong, Yanghui Liu +3 · 6 citations
Economics, Econometrics and Finance · Mathematics · Social Sciences · #Applied mathematics #Backward Euler method #Brownian motion #Computer science #Convergence (economics) #Differential equation #Euler equations #Euler method #Euler's formula #FOS: Mathematics #Financial Risk and Volatility Modeling #Fractional Brownian motion #Insurance, Mortality, Demography, Risk Management #Malliavin calculus #Mathematical analysis #Mathematics #Probability (math.PR) #Rate of convergence #Scheme (mathematics) #Stochastic differential equation #Stochastic partial differential equation #Stochastic processes and financial applications #Weak convergence #math.PR
paper · pdf · doi:10.48550/arxiv.1306.1458
published in arXiv (Cornell University) (Cornell University) · This paper has been withdrawn. It has been replaced an updated version of the paper: arXiv:1408.6471
openalex publication_date 2013/06/06 · openalex created_date 2016/06/24 · arxiv created 2017/03/04 · arxiv updated 2017/03/07 · openalex updated_date 2026/07/28
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.