JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2026, Vol. 61 ›› Issue (9): 64-73.doi: 10.6040/j.issn.1671-9352.0.2024.345

Previous Articles    

Existence and smoothness for multiple local time of fractional Brownian motion

ZHANG Cuiyun, GUO Jingjun, HU Feng   

  1. School of Statistics and Data Science, Lanzhou University of Finance and Economics, Lanzhou 730020, Gansu, China
  • Published:2026-09-30

PDF (PC)

10

Abstract: The Wiener chaos decomposition for the multiple local time of one-dimensional fractional Brownian motion is established. Its existence and smoothness are proved. The relevant results are extended to the d-dimensional fractional Brownian motion for d≥2. When x=0, the Wiener chaos decomposition and smoothness for the multiple local time of d-dimensional fractional Brownian motion are obtained.

Key words: fractional Brownian motion, local time, Wiener chaos decomposition, smoothness

CLC Number: 

  • O211
[1] Guo Jingjun, Zhang Cuiyun, Ma Aiqin. Derivative of multiple self-intersection local time for fractional Brownian motion[J]. Journal of Theoretical Probability, 2024, 37(1): 623-641.
[2] 张翠芸, 郭精军, 马爱琴. 对称Stable过程的多重自相交局部时研究[J]. 数学学报(中文版), 2023, 66(4): 779-790. Zhang Cuiyun, Guo Jingjun, Ma Aiqin. Study on the multiple self-intersection local time of symmetric stable processes[J]. Acta Mathematica Sinica(Chinese Series), 2023, 66(4): 779-790.
[3] 桑利恒, 陈振龙, 郝晓珍. 双分数布朗运动重整化自相交局部时的光滑性[J]. 数学物理学报, 2020, 40(3): 796-810. Sang Liheng, Chen Zhenlong, Hao Xiaozhen. Smoothness for the renormalized self-intersection local time of bifractional Brownian motion[J]. Acta Mathematica Scientia, 2020, 40(3): 796-810.
[4] Rosen J, Yor M. Tanaka formulae and renormalization for triple intersections of Brownian motion in the plane[J]. The Annals of Probability, 1991, 19(1): 142-159.
[5] Imkeller P, Yan J A. Multiple intersection local time of planar Brownian motion as a particular Hida distribution[J]. Journal of Functional Analysis, 1996, 140(1): 256-273.
[6] Bornales J, Oliveira M J, Streit L. Chaos decomposition and gap renormalization of Brownian self-intersection local times[J]. Reports on Mathematical Physics, 2016, 77(2): 141-152.
[7] Faria M, Drumond C, Streit L. The renormalization of self-intersection local times Ⅰ: the chaos expansion[J]. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2011, 3(2): 223-236.
[8] Bass R F, Khoshnevisan D. Intersection local times and Tanaka formulas[J]. Annales de lIHP Probabilités et Statistiques, 1993, 29(3): 419-451.
[9] Shieh N R. White noise analysis and Tanaka formula for intersections of planar Brownian motion[J]. Nagoya Mathematical Journal, 1991, 122(1): 1-17.
[10] Mendonça S, Streit L. Multiple intersection local times in terms of white noise[J]. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2001, 4(4): 533-543.
[11] Nualart D, Vives J. Chaos expansions and local times[J]. PublicacionsMatematiques, 1992, 36(2B): 827-836.
[12] Imkeller P, Perez-Abreu V, Vives J. Chaos expansions of double intersection local time of Brownian motion in Rd and renormalization[J]. Stochastic Processes and their Applications, 1995, 56(1): 1-34.
[13] Wang Xiangjun, Guo Jingjun, Jiang Guo. Collision local times of two independent fractional Brownian motions[J]. Frontiers of Mathematics in China, 2011, 6(2): 325-338.
[14] Rosen J. The intersection local time of fractional Brownian motion in the plane[J]. Journal of Multivariate Analysis, 1987, 23(1): 37-46.
[15] Hu Y, Nualart D. Renormalized self-intersection local time for fractional Brownian motion[J]. The Annals of Probability, 2005, 33(3): 948-983.
[16] Yu Xianye. Smoothness of self-intersection local time of multidimensional fractional Brownian motion[J]. Communications in Statistics: Theory and Methods, 2019, 48(17): 4278-4293.
[17] Jung P, Markowsky G. Hölder continuity and occupation-time formulas for fBm self-intersection local time and its derivative[J]. Journal of Theoretical Probability, 2015, 28(1): 299-312.
[18] Oliveira M J, Da Silva J L, Streit L. Intersection local times of independent fractional Brownian motions as generalized white noise functionals[J]. Acta Applicandae Mathematicae, 2011, 113(4): 17-39.
[19] Jiang Yiming, Wang Yongjin. On the collision local time of fractional Brownian motions[J]. Chinese Annals of Mathematics(Series B), 2007, 28(3): 311-320.
[20] Wu Dongsheng, Xiao Yimin. Regularity of intersection local times of fractional Brownian motions[J]. Journal of Theoretical Probability, 2010, 23(4): 972-1001.
[21] Eddahbi M, Lacayo R, Soléj L, et al. Regularity of the local time for the d-dimensional fractional Brownian motion with N-parameters[J]. Stochastic Analysis and Applications, 2005, 23(2): 383-400.
[22] Nualart D. The Malliavin calculus and related topics[M]. Berlin: Springer, 2006: 321-349.
[23] Imkeller P, Weisz F. The asymptotic behaviour of local times and occupation integrals of the N-parameter Wiener process in Rd[J]. Probability Theory and Related Fields, 1994, 98(1): 47-75.
[1] ZHANG Zhaoliu, FAN Xiaoming. European option pricing under double Heston jump-diffusion model with generalized fractional Brownian motion [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2025, 60(3): 60-68.
[2] Yaru ZHANG,Li XIA,Dianqiu ZHANG. Perpetual American lookback option pricing under mixed bi-fractional Brownian motion [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2024, 59(4): 98-107.
[3] Cuiyun ZHANG,Jingjun GUO,Aiqin MA. Parameter estimation for the sub-fractional Vasicek model based on discrete observation [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(11): 15-26.
[4] GUO Jing-jun, WANG Yu-bing, BAI Ya-nan. American option pricing and simulation under the mixed fractional Heston-CIR model [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(9): 46-54.
[5] AN Xiang, GUO Jing-jun. Pricing and simulation of lookback options under the mixed sub-fractional jump-diffusion model [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2022, 57(4): 100-110.
[6] MA Wei-feng, CHEN Peng-yu. Solution manifold and its C1-smoothness for differential equations with state-dependent delay [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(2): 92-96.
[7] PENG Bo, GUO Jing-jun. Asset pricing and simulation under the environment of jumping and mixed Gaussian process [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(5): 105-113.
[8] CUI Jing, LIANG Qiu-ju. Existence and controllability of nonlocal stochastcic integro-differential equations driven by fractional Brownian motion [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(12): 81-88.
[9] DENG Long-juan, ZHU Dong-jin, SHEN Guang-jun. Power variation of weighted-fractional Brownian motion and application [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(06): 19-26.
[10] YANG Zhao-qiang. A kind of European lookback option pricing model under fractional  jump-diffusion mixed fractional Brownian motion [J]. J4, 2013, 48(6): 67-74.
[11] SHEN Ming-xuan1, HE Chao-lin2. Geometric average asian option pricing in fractional brownian environment [J]. J4, 2013, 48(3): 48-52.
[12] YANG Zhao-qiang. A kind European lookback option pricing model in mixed fractional Brownian motion environment [J]. J4, 2012, 47(9): 105-109.
[13] SHEN Guang-jun 1,2, HE Kun 3, YAN Li-tan 3. Remarks on sub-fractional Brownian motion [J]. J4, 2011, 46(3): 102-108.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd