ε-Strong Simulation of Fractional Brownian Motion and Related Stochastic Differential Equations

• 成果类型: Article
• 署名作者: Chen, Yi; Dong, Jing; Ni, Hao
• 署名单位: Northwestern University; Columbia University; University of London; University College London
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2020.1078
• 发表日期: 2021
• 页码: 559-594
• 关键词: rough path; driven

摘要

Consider a fractional Brownian motion (fBM) B-H = {B-H(t) : t is an element of [0, 1]} with Hurst index H is an element of (0, 1). We construct a probability space supporting both B-H and a fully simulatable process (B) over cap (H)(epsilon) such that sup(t is an element of[0,1])vertical bar B-H(t) - (B) over cap (H)(epsilon)(t)vertical bar <= epsilon with probability one for any user-specified error bound epsilon > 0. When H > 1/2, we further enhance our error guarantee to the alpha-Holder norm for any alpha is an element of (1/2, H). This enables us to extend our algorithm to the simulation of fBM-driven stochastic differential equations Y = {Y(t) : t is an element of [0, 1]}. Under mild regularity conditions on the drift and diffusion coefficients of Y, we construct a probability space supporting both Y and a fully simulatable process (Y) over cap (epsilon) such that sup(t is an element of[0,1])vertical bar Y(t) - (Y) over cap (epsilon)(t)vertical bar <= epsilon with probability one. Our algorithms enjoy the tolerance-enforcement feature, under which the error bounds can be updated sequentially in an efficient way. Thus, the algorithms can be readily combined with other advanced simulation techniques to estimate the expectations of functionals of fBMs efficiently.