Tanaka Formula for Multidimensional Brownian Motions

We study the Tanaka formula for multidimensional Brownian motions in the framework of generalized Wiener functionals. More precisely, we show that the submartingale U(B _t –x) is decomposed in the sence of generalized Wiener functionals into the sum of a martingale and the Brownian local time, U being twice of the kernel of Newtonian potential and B _t the multidimensional Brownian motion. We also discuss on an aspect of the Tanaka formula for multidimensional Brownian motions as the Doob–Meyer decomposition.     

On the Tanaka formula for the derivative of self-intersection local time of fractional Brownian motion

The derivative of self-intersection local time (DSLT) for Brownian motion was introduced by Rosen (2005) and subsequently used by others to study the L²$L^2$ and L³$L^3$ moduli of continuity of Brownian local time. A version of the DSLT for fractional Brownian motion (fBm) was introduced in Yan et al. (2008); however, the definition given there presents difficulties, since it is motivated by an incorrect application of the fractional Itô formula. To rectify this, we introduce a modified DSLT for fBm and prove existence using an explicit Wiener chaos expansion. We will then argue that our modification is the natural version of the DSLT by rigorously proving the corresponding Tanaka formula. This formula corrects a formal identity given in both Rosen (2005) and Yan et al. (2008). In the course of this endeavor we prove a Fubini theorem for integrals with respect to fBm. The Fubini theorem may be of independent interest, as it generalizes (to Hida distributions) similar results previously seen in the literature. As a further byproduct of our investigation, we also provide a small correction to an important technical second-moment bound for fBm which has appeared in the literature many times.     

Intersection Local Time for Two Independent Fractional Brownian Motions

Let B ^H and be two independent, d-dimensional fractional Brownian motions with Hurst parameter H∈(0,1). Assume d≥2. We prove that the intersection local time of B ^H and

A Note on the Local Time of Fractional Brownian Motion

Asymptotic behavior of the local time at the origin of q-dimensional fractional Brownian motion is considered when the index approaches the critical value 1/q. It is proved that, under a suitable (temporally inhomogeneous) normalization, it converges in law to the inverse of an extremal process which appears in the extreme value theory.     

Stratonovich Calculus with Respect to Fractional Brownian Sheet

Abstract

We introduce two types of Stratonovich stochastic integrals for two-parameter process. The relationship of Stratonovich integrals to Skorohod integrals will be investigated. By using this relationship, we prove that a differentiation formula for fractional Brownian sheet in Stratonovich form can be expressed as the sum of Stratonovich integrals of two types introduced in this article.     

On the Collision Local Time of Fractional Brownian Motions

In this paper, the existence and smoothness of the collision local time are proved for two independent fractional Brownian motions, through L2 convergence and Chaos expansion. Furthermore, the regularity of the collision local time process is studied.     

次分数布朗运动局部时的研究

设X^H={X^H(t),t∈R+}是一个取值于R^d参数为H的次分数布朗运动.本文给出了X^H在单参数情况下局部时的Holder条件和尾概率估计.同时,还给出了X^H在多参数情况下局部时的存在性及L^2表示.     

分式Brown运动的局部时:白噪声分析法

本文利用白噪声分析的方法,讨论了分式布朗运动的局部时,即将其看作一个Hida分布.进一步,给出分式布朗运动的局部时的混沌分解以及局部时平方可积性.     

Fractional Brownian Density Process and Its Self-Intersection Local Time of Order k

The fractional Brownian density process is a continuous centered Gaussian ( ^d )-valued process which arises as a high-density fluctuation limit of a Poisson system of independent d-dimensional fractional Brownian motions with Hurst parameter H. ( ( ^d ) is the space of tempered distributions). The main result proved in the paper is that if the intensity measure of the (initial) Poisson random measure on ^d is either the Lebesgue measure or a finite measure, then the density process has self-intersection local time of order k 2 if and only if Hd < k/(k – 1). The latter is also the necessary and sufficient condition for existence of multiple points of order k for d-dimensional fractional Brownian motion, as proved by Talagrand¹². This result extends to a non-Markovian case the relationship known for (Markovian) symmetric -stable Lévy processes and their corresponding density processes. New methods are used in order to overcome the lack of Markov property. Other properties of the fractional Brownian density process are also given, in particular the non-semimartingale property in the case H 1/2, which is obtained by a general criterion for the non-semimartingale property of real Gaussian processes that we also prove.     

Fractional Brownian Motion and Sheet as White Noise Functionals

In this short note, we show that it is more natural to look the fractional Brownian motion as functionals of the standard white noises, and the fractional white noise calculus developed by Hu and Фksendal follows directly from the classical white noise functional calculus. As examples we prove that the fractional Girsanov formula, the Ito type integrals and the fractional Black-Scholes formula are easy consequences of their classical counterparts. An extension to the fractional Brownian sheet is also briefly discussed.     

Continuity in Law with Respect to the Hurst Parameter of the Local Time of the Fractional Brownian Motion

We give a result of stability in law of the local time of the fractional Brownian motion with respect to small perturbations of the Hurst parameter. Concretely, we prove that the law (in the space of continuous functions) of the local time of the fractional Brownian motion with Hurst parameter H converges weakly to that of the local time of , when H tends to H ₀.      

Stochastic Differential Equations Driven by Fractional Brownian Motion and Standard Brownian Motion

Abstract

We prove an existence and uniqueness theorem for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter H > 1/2 and a multidimensional standard Brownian motion. The proof relies on some a priori estimates, which are obtained using the methods of fractional integration and the classical Itô stochastic calculus. The existence result is based on the Yamada–Watanabe theorem.     

利用Ito公式求布朗运动和几何布朗运动的矩

利用Ito公式及Ito积分的性质求出了布朗运动和几何布朗运动的矩的一般形式,同时指出可以利用这种方法求其他扩散过程的矩.     

Stochastic Analysis of the Fractional Brownian Motion

Since the fractional Brownian motion is not a semi-martingale, the usual Ito calculus cannot be used to define a full stochastic calculus. However, in this work, we obtain the Itô formula, the Itô–Clark representation formula and the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations.     

Maximal Inequalities for Fractional Brownian Motion: An Overview

We give an overview of some maximal inequalities and limit theorems for the tail probabilities for the supremum of a fractional Brownian motion.     

Integral Representation with Adapted Continuous Integrand with Respect to Fractional Brownian Motion

We show that if a random variable is a final value of an adapted Hölder continuous process, then it can be represented as a stochastic integral with respect to fractional Brownian motion, and the integrand is an adapted process, continuous up to the final point.     

White noise-based stochastic calculus with respect to multifractional Brownian motion

Stochastic calculus with respect to fractional Brownian motion (fBm) has attracted a lot of interest in recent years, motivated in particular by applications in finance and Internet traffic modeling. Multifractional Brownian motion (mBm) is an extension of fBm enabling to control the local regularity of the process. It is obtained by replacing the constant Hurst parameter H of fBm by a function h(t), thus allowing for a finer modelling of various phenomena.

In this work we extend to mBm the construction of the Wick–Itô stochastic integral with respect to fBm, as originally proposed in Bender (Stoch. Process. Appl. 104 (2003), pp. 81–106), Bender (Bernouilli 9(6) (2003), pp. 955–983), Biagini et al. (Proceedings of Royal Society, special issue on stochastic analysis and applications, 2004, pp. 347–372) and Elliott and Van der Hoek (Math. Finance 13(2) (2003), pp. 301–330). In that view, a multifractional white noise is defined and used to integrate with respect to mBm a large class of stochastic processes using Wick products. Itô formulas (both for tempered distributions and for functions with sub-exponential growth) are obtained, as well as a Tanaka Formula.     

Estimation for Translation of a Process Driven by Fractional Brownian Motion

Abstract

We investigate the general problem of estimating the translation of a stochastic process governed by a stochastic differential equation driven by a fractional Brownian motion. The special case of the Ornstein-Uhlenbeck process is discussed in particular.