Some properties of the sub-fractional Brownian motion

We study several properties of the sub-fractional Brownian motion (fBm) introduced by Bojdecki et al. related to those of the fBm. This process is a self-similar Gaussian process depending on a parameter H ∈ (0, 2) with non stationary increments and is a generalization of the Brownian motion (Bm).

The strong variation of the indefinite stochastic integral with respect to sub-fBm is also discussed.     

The existence and uniqueness of the solution of an integral equation driven by a p-semimartingale of special type

We consider the integral equation driven by a standard Brownian motion and fractional Brownian motion (fBm). Since fBm is not a semimartingale, we cannot use the semimartingale theory to define an integral with respect to the fBm. Furthermore, a well-developed theory of stochastic differential equations is not applicable to solve it. Existence and uniqueness conditions are obtained for a solution in the space of continuous functions with q-bounded variation, q>2.     

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.     

On the Weighted Local Time and the Tanaka Formula for the Multidimensional Fractional Brownian Motion

Abstract

We determine the weighted local time for the multidimensional fractional Brownian motion from the occupation time formula. We also discuss on the Itô and Tanaka formula for the multidimensional fractional Brownian motion. In these formulas the Skorohod integral is applicable if the Hurst parameter of fractional Brownian motion is greater than 1/2. If the Hurst parameter is less than 1/2, then we use the Skorohod type integral introduced by Nualart and Zakai for the stochastic integral and establish the Itô and Tanaka formulas.     

Covariance of stochastic integrals with respect to fractional Brownian motion

We find an explicit expression for the cross-covariance between stochastic integral processes with respect to a $d$-dimensional fractional Brownian motion (fBm) $B_t$ with Hurst parameter $H>\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, where the integrands are vector fields applied to $B_t$. It provides, for example, a direct alternative proof of Y. Hu and D. Nualart’s result that the stochastic integral component in the fractional Bessel process decomposition is not itself a fractional Brownian motion.     

Stochastic integration with respect to multifractional Brownian motion via tangent fractional Brownian motions

Stochastic integration w.r.t. fractional Brownian motion (fBm) has raised strong interest in recent years, motivated in particular by applications in finance and Internet traffic modelling. Since fBm is not a semi-martingale, stochastic integration requires specific developments. Multifractional Brownian motion (mBm) generalizes fBm by letting the local Hölder exponent vary in time. This is useful in various areas, including financial modelling and biomedicine. The aim of this work is twofold: first, we prove that an mBm may be approximated in law by a sequence of “tangent” fBms. Second, using this approximation, we show how to construct stochastic integrals w.r.t. mBm by “transporting” corresponding integrals w.r.t. fBm. We illustrate our method on examples such as the Wick–Itô, Skorohod and pathwise integrals.     

Non-densely defined impulsive neutral stochastic functional differential equations driven by fBm in Hilbert space with infinite delay

We study a class of non-densely defined impulsive neutral stochastic functional differential equations driven by an independent cylindrical fractional Brownian motion (fBm) with Hurst parameter H∈ (1/2, 1) in the Hilbert space. We prove the existence and uniqueness of the integral solution for this kind of equations with the coefficients satisfying some non-Lipschitz conditions. The results are obtained by using the method of successive approximation.     

Convergence of delay differential equations driven by fractional Brownian motion

In this note, we prove an existence and uniqueness result of solution for stochastic differential delay equations with hereditary drift driven by a fractional Brownian motion with Hurst parameter H > 1/2. Then, we show that, when the delay goes to zero, the solutions to these equations converge, almost surely and in L ^p , to the solution for the equation without delay. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann–Stieltjes integral.     

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.     

Random variables as pathwise integrals with respect to fractional Brownian motion

We give both necessary and sufficient conditions for a random variable to be represented as a pathwise stochastic integral with respect to fractional Brownian motion with an adapted integrand. We also show that any random variable is a value of such integral in an improper sense and that such integral can have any prescribed distribution. We discuss some applications of these results, in particular, to fractional Black–Scholes model of financial market.     

The Wiener integral with respect to second order processes with stationary increments

We prove that for any second order stochastic process X with stationary increments with continuous paths and continuous variance function, there exists a tempered measure μ (for which we give an explicit expression) related with the domain of the Wiener integral with respect to X as follows: the space of tempered distributions f such that the Fourier transform of f is square integrable with respect to μ is always a dense subset of the domain of the Wiener integral. Moreover, we provide sufficient conditions on μ in order that the domain of the integral is exactly this space of distributions. We apply our results to the fractional Brownian motion. In particular, it is proved that the domain of the Wiener integral with respect to the fractional Brownian motion with Hurst parameter H>1/2 contains distributions that are not given by locally integrable functions, this fact was suggested by Pipiras and Taqqu (2000) in [5]. We have also considered the example of the process given by Ornstein and Uhlenbeck as a model for the position of a Brownian particle.     

On a ogawa–type integral with application to the fractional brownian motion

We construct a deterministic Ogawa–type integral with respect to a continuous function that, in particular, can be a trajectory of the Fractional Brownian motion. This integral is related with the Stratonovich integral and with the integrals introduced by Ciesielski et altri and Zähle. We give a sufficient condition for the integrability of a function in this sense, that does not imply its continuity. Under this sufficient condition, we obtain a Besov regularity property of the indefinite integral. We also study the stochastic Ogawa integral for stochastic processes when integrate with respect to the Fractional Brownian motion of Hurst parameter H ∈ (1/2, 1)     

Limites approximatives sur l'espace de Wiener

We use a stochastic integral which was first constructed by Nualart, Zakai and Ogawa, to show, for the variables of the second Wiener chaos, that the existence of this integral imply that these variables possess an approximate limit with respect to measurable norms defined by Gross. Moreover, this limit does not depend on the choice of the norm. Furthermore, we show that measurable norms possess an approximate limit with respect to quadratic norms. The main argument is a correlation inequality proved by Hargé.     

Besov regularity for the indefinite Skorohod integral with respect to the fractional Brownian motion: The singular case

Using the techniques of the Malliavin calculus and the properties of Gaussian processes, we prove that the paths of the indefinite Skorohod integral with respect to the fractional Brownian motion (fBm) with Hurst parameter less than 1/2 belongs to the Besov space B p , X H , for any p >(1/ H ).     

A family of integral representations for the Brownian variables

The natural filtration of a real Brownian motion and its excursion filtration are sharing a fundamental property: the property of integral representation. As a consequence, every Brownian variable admits two distinct integral representations. We show here that there are other integral representations of the Brownian variables. They make use of a stochastic flow studied by Bass and Burdzy. Our arguments are inspired by Rogers and Walsh's results on stochastic integration with respect to the Brownian local times.     

Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter

We define a stochastic integral with respect to fractional Brownian motion B^H with Hurst parameter that extends the divergence integral from Malliavin calculus. For this extended divergence integral we prove a Fubini theorem and establish versions of the formulas of Itô and Tanaka that hold for all . Then we use the extended divergence integral to show that for every and all , the Russo–Vallois symmetric integral exists and is equal to , where G^′=g, while for , does not exist.     

多重分数斯特拉托诺维奇积分的一个注记

本文研究了多重分数斯特拉托诺维奇积分,通过卷积逼近技巧和分数布朗运动的随机积分的性质,构造了当Hurst参数小于二分之一时的多重随机积分.这种方法是新的不同于文[8]中的构造方法.     

Random homogenization and convergence to integrals with respect to the Rosenblatt process

This paper concerns the random fluctuation theory of a one dimensional elliptic equation with highly oscillatory random coefficient. Theoretical studies show that the rescaled random corrector converges in distribution to a stochastic integral with respect to Brownian motion when the random coefficient has short-range correlation. When the random coefficient has long-range correlation, it was shown for a large class of random processes that the random corrector converged to a stochastic integral with respect to fractional Brownian motion. In this paper, we construct a class of random coefficients for which the random corrector converges to a non-Gaussian limit. More precisely, for this class of random coefficients with long-range correlation, the properly rescaled corrector converges in distribution to a stochastic integral with respect to a Rosenblatt process.     

Existence of solutions for fractional neutral functional differential equations driven by fBm with infinite delay

In this paper, we consider a class of fractional neutral stochastic functional differential equations with infinite delay driven by a cylindrical fractional Brownian motion (fBm) in a real separable Hilbert space. We prove the existence of mild solutions by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.     

Asymptotic distributions of solutions of ordinary differential equations with wide band noise inputs: approximate invariant measures

Systems of Wick stochastic differential equations are studied. Using an estimate on the Wick product we apply Picard iteration to prove a general existence and uniqueness theorem for systems of Wick stochastic differential equations. We also show the solution is stable with respect to perturbations of the noise. This result is used to show that the solution of a linear system of Wick stochastic differential equations driven by smoothed Brownian motion tends to the solution of the corresponding It equation as the smoothed process tends to Brownian motion