scholarly journals The generalized quadratic covariation for fractional Brownian motion with Hurst index less than 1/2

2014 ◽  
Vol 17 (04) ◽  
pp. 1450030 ◽  
Author(s):  
Litan Yan ◽  
Junfeng Liu ◽  
Chao Chen
Keyword(s):  
Banach Space ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Local Time ◽  
Borel Function ◽  
Time Dependent ◽  
Hurst Index ◽  
Dependent Case ◽  
Measurable Functions ◽  
Quadratic Covariation

In this paper, we study the generalized quadratic covariation of f(BH) and BH defined by [Formula: see text] in probability, where f is a Borel function and BH is a fractional Brownian motion with Hurst index 0 < H < 1/2. We construct a Banach space [Formula: see text] of measurable functions such that the generalized quadratic covariation exists in L2(Ω) and the Bouleau–Yor identity takes the form [Formula: see text] provided [Formula: see text], where [Formula: see text] is the weighted local time of BH. These are also extended to the time-dependent case, and as an application we give the identity between the generalized quadratic covariation and the 4-covariation [g(BH), BH, BH, BH] when [Formula: see text].

scholarly journals The quadratic covariation for a weighted fractional Brownian motion

2017 ◽  
Vol 17 (04) ◽  
pp. 1750029
Author(s):  
Xichao Sun ◽  
Litan Yan ◽  
Qinghua Zhang
Keyword(s):  
Banach Space ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Asymptotic Property ◽  
Local Time ◽  
Beta Function ◽  
Measurable Functions ◽  
Quadratic Covariation

Let [Formula: see text] be a weighted fractional Brownian motion with indices [Formula: see text] and [Formula: see text] satisfying [Formula: see text] [Formula: see text] [Formula: see text]. In this paper, motivated by the asymptotic property [Formula: see text] for all [Formula: see text], we consider the generalized quadratic covariation [Formula: see text] defined by [Formula: see text] provided the limit exists uniformly in probability. We construct a Banach space [Formula: see text] of measurable functions such that the generalized quadratic covariation exists in [Formula: see text] and the generalized Bouleau–Yor identity [Formula: see text] holds for all [Formula: see text], where [Formula: see text] is the weighted local time of [Formula: see text] and [Formula: see text] is the Beta function.

scholarly journals A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

2014 ◽  
Vol 51 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Dawei Hong ◽  
Shushuang Man ◽  
Jean-Camille Birget ◽  
Desmond S. Lun
Keyword(s):  
Brownian Motion ◽  
Convergence Rate ◽  
Fractional Brownian Motion ◽  
Parallel Algorithm ◽  
Uniform Approximation ◽  
Haar Wavelets ◽  
Hurst Index ◽  
Sample Paths ◽  
Vanishing Moment ◽  
Uniform Expansion

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

scholarly journals Second-order limit laws for occupation times of fractional Brownian motion

2017 ◽  
Vol 54 (2) ◽  
pp. 444-461 ◽  
Author(s):  
Fangjun Xu
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Method Of Moments ◽  
Second Order ◽  
Hurst Index ◽  
Occupation Times ◽  
Limit Laws ◽  
Additive Functionals ◽  
Finite Dimensional ◽  
Functional Version

Abstract We prove a second-order limit law for additive functionals of a d-dimensional fractional Brownian motion with Hurst index H = 1 / d, using the method of moments and extending the Kallianpur–Robbins law, and then give a functional version of this result. That is, we generalize it to the convergence of the finite-dimensional distributions for corresponding stochastic processes.

scholarly journals Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion With Hurst Index Large Than 0.5 I: Self-Repelling Case

2022 ◽  
Vol 9 ◽  
Author(s):  
Han Gao ◽  
Rui Guo ◽  
Yang Jin ◽  
Litan Yan
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Large Time ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Hurst Index ◽  
Interacting Diffusion ◽  
Two Parameters

Let SH be a sub-fractional Brownian motion with index 12<H<1. In this paper, we consider the linear self-interacting diffusion driven by SH, which is the solution to the equationdXtH=dStH−θ(∫0tXtH−XsHds)dt+νdt,X0H=0,where θ &lt; 0 and ν∈R are two parameters. Such process XH is called self-repelling and it is an analogue of the linear self-attracting diffusion [Cranston and Le Jan, Math. Ann. 303 (1995), 87–93]. Our main aim is to study the large time behaviors. We show the solution XH diverges to infinity, as t tends to infinity, and obtain the speed at which the process XH diverges to infinity as t tends to infinity.

scholarly journals Nonparametric Regression with Subfractional Brownian Motion via Malliavin Calculus

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Yuquan Cang ◽  
Junfeng Liu ◽  
Yan Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Nonparametric Regression ◽  
Local Time ◽  
Kernel Function ◽  
Malliavin Calculus ◽  
Bandwidth Parameter ◽  
Subfractional Brownian Motion ◽  
Normal Law

We study the asymptotic behavior of the sequenceSn=∑i=0n-1K(nαSiH1)(Si+1H2-SiH2),asntends to infinity, whereSH1andSH2are two independent subfractional Brownian motions with indicesH1andH2, respectively.Kis a kernel function and the bandwidth parameterαsatisfies some hypotheses in terms ofH1andH2. Its limiting distribution is a mixed normal law involving the local time of the sub-fractional Brownian motionSH1. We mainly use the techniques of Malliavin calculus with respect to sub-fractional Brownian motion.

Sign in / Sign up

Export Citation Format

Share Document