An integral functional driven by fractional Brownian motion

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:129

Article

Sun, Xichao¹  Yan, Litan²  Yu, Xianye³ 

[1] Bengbu Univ, Coll Sci, Dept Math, 1866 Caoshan Rd, Bengbu 233030, Peoples R China

[2] Donghua Univ, Coll Sci, Dept Math, 2999 North Renmin Rd, Shanghai 201620, Peoples R China

[3] Sun Yat Sen Univ, Sch Math, 135 Xingang Xi Rd, Guangzhou 510275, Guangdong, Peoples R China

关键词: Fractional Brownian motion;    Malliavin calculus;    Local time;    Fractional Ito formula;    Cauchy's principal value;   

DOI  :  10.1016/j.spa.2018.07.004

来源: Elsevier

【 摘 要 】

Let B-H be a fractional Brownian motion with Hurst index 0 < H < 1 and the weighted local time L-H (., t). In this paper, we consider the integral process C-t(H) (a) := lim (epsilon down arrow 0) integral(t)(0) 1 ({vertical bar BsH -a vertical bar >=epsilon}) 2Hs(2H-1)/B-s(H )- a ds -HLH (., t)(a), t >= 0 in L-2 (Omega) with a is an element of R, where H denotes the Hilbert transform. We show that the Skorohod integral integral(.)(0) log vertical bar B-s(H) - a vertical bar d B-s(H) exists in L-2 (Omega) and the fractional Yamada formula (B-t(H) - a) log vertical bar B-t(H) - a vertical bar - B-t(H) + a log vertical bar a vertical bar - integral(t)(0) log vertical bar B-s(H) - a vertical bar d B-s(H) = 1/2 C-t(H) (a) holds for all a is an element of R, t >= 0. Moreover, we introduce the next occupation type formula: integral(R) C-t(H) (a)g(a)da = 2H integral(t)(0) (Hg)(B-s(H))s(2H-1)ds for all continuous functions g with compact support. (C) 2018 Elsevier B.V. All rights reserved.

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