An integral functional driven by fractional Brownian motion

作者：Sun, Xichao; Yan, Litan^*; Yu, Xianye

来源：Stochastic Processes and Their Applications, 2019, 129(7): 2249-2285.

DOI：10.1016/j.spa.2018.07.004

• Fractional Brownian motion
• Malliavin calculus
• Local time
• Fractional Ito formula
• Cauchy's principal value

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.

• 出版日期2019-7
• 单位东华大学; 蚌埠学院; 中山大学