RATE OF CONVERGENCE AND ASYMPTOTIC ERROR DISTRIBUTION OF EULER APPROXIMATION SCHEMES FOR FRACTIONAL DIFFUSIONS

• 成果类型: Article
• 署名作者: Hu, Yaozhong; Liu, Yanghui; Nualart, David
• 署名单位: University of Kansas
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/15-AAP1114
• 发表日期: 2016
• 页码: 1147-1207
• 关键词: differential-equations driven; LIMIT-THEOREMS; sdes driven; integration; INEQUALITY; calculus; Respect

摘要

For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter H > 1/2, it is known that the existing (naive) Euler scheme has the rate of convergence n(1-2H). Since the limit H -> 1/2 of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for Ito SDEs for H = 1/2, the convergence rate of the naive Euler scheme deteriorates for H -> 1/2. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for H = 1/2, and it has the rate of convergence gamma(-1)(n), where gamma(n) = n(2H-1/2) when H < 3/4, gamma(n) = n/root log n when H = 3/4 and gamma(n) = n if H > 3/4. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if {X-t, 0 <= t <= T} is the solution of a SDE driven by a fBm and if {X-t(n), 0 <= t <= T} is its approximation obtained by the new modified Euler scheme, then we prove that gamma(n) (X-n - X) converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when H is an element of (1/2, 3/4]. In the case H > 3/4, we show the L-p convergence of n(X-t(n) - X-t), and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme.