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.     

On a Multiple Stratonovich-type Integral for Some Gaussian Processes

We construct a multiple Stratonovich-type integral with respect to Gaussian processes with covariance function of bounded variation. This construction is based on the previous definition of the multiple Itô-type integral given by Huang and Cambanis [Ann. Propab. 6(4), 585–614] and on a Hu–Meyer formula (that is, an expression of the multiple Stratonovich integral as a sum of Itô-type integrals of inferior or equal order) for the elementary functions. We also apply our results to the fractional Brownian motion with Hurst parameter $H > \frac{1}{2}We construct a multiple Stratonovich-type integral with respect to Gaussian processes with covariance function of bounded variation. This construction is based on the previous definition of the multiple It?-type integral given by Huang and Cambanis [Ann. Propab. 6(4), 585–614] and on a Hu–Meyer formula (that is, an expression of the multiple Stratonovich integral as a sum of It?-type integrals of inferior or equal order) for the elementary functions. We also apply our results to the fractional Brownian motion with Hurst parameter .     

Wick–Itô formula for regular processes and applications to the Black and Scholes formula

We derive a Wick–Itô formula, that is, an Itô-type formula based on Wick integration. We derive it in the context of regular Gaussian processes which include Brownian motion and fractional Brownian motion with Hurst parameter greater than 1/2. We then consider applications to the Black and Scholes formula for the pricing of a European call option. It has been shown that using Wick integration in this context is problematic for economic reasons. We show that it is also problematic for mathematical reasons because the resulting Black and Scholes formula depends only on the variance of the process and not on its dependence structure.     

An Itô Formula of Generalized Functionals and Local Time for Fractional Brownian Sheet

Abstract

By using the white noise theory for a fractional Brownian sheet, we derive an Itô formula for the generalized functionals for the fractional Brownian sheet with arbitrary Hurst parameters H ₁, H ₂ ∈ (0,1). As an application, we give the integral representations for two versions of local times of a fractional Brownian sheet, respectively.     

Wick-Itô Formula for Gaussian Processes

Abstract

A Wick-Itô formula for Gaussian processes is obtained. This is a change of variables formula, which is to Wick-Itô integrals what the usual Itô formula is to Itô integrals. The conditions are weak enough to allow processes with infinite quadratic variation. They are satisfied by fractional Brownian motion with parameter 1/4 < H < 1.     

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.     

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 ).     

Skorohod integration and stochastic calculus beyond the fractional Brownian scale

We extend the Skorohod integral, allowing integration with respect to Gaussian processes that can be more irregular than any fractional Brownian motion. This is done by restricting the class of test random variables used to define Skorohod integrability. A detailed analysis of the size of this class is given; it is proved to be non-empty even for Gaussian processes which are not continuous on any closed interval. Despite the extreme irregularity of these stochastic integrators, the Skorohod integral is shown to be uniquely defined, and to be useful: an Ito formula is established; it is employed to derive a Tanaka formula for a corresponding local time; linear additive and multiplicative stochastic differential equations are solved; an analysis of existence for the stochastic heat equation is given.     

Linear Stochastic Differential Equations Driven by a Fractional Brownian Motion with Hurst Parameter Less than 1/2

Abstract

In this article, we use the chaos decomposition approach to establish the existence of a unique continuous solution to linear fractional differential equations of the Skorohod type. Here, the coefficients are deterministic, the initial condition is anticipating and the underlying fractional Brownian motion has Hurst parameter less than 1/2. We provide an explicit expression for the chaos decomposition of the solution in order to show our results.     

Solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian Motions

The local existence and uniqueness of the solutions to backward stochastic differential equations(BSDEs, in short) driven by both fractional Brownian motions with Hurst parameter H ∈(1/2, 1) and the underlying standard Brownian motions are studied. The generalization of the It formula involving the fractional and standard Brownian motions is provided. By theory of Malliavin calculus and contraction mapping principle, the local existence and uniqueness of the solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian motions are obtained.     

Numerical Solution Based on Hat Functions for Solving Nonlinear Stochastic Itô Volterra Integral Equations Driven by Fractional Brownian Motion

This paper presents a numerical method for solving nonlinear stochastic Itô Volterra integral equations driven by fractional Brownian motion with Hurst parameter \( H \in (0,1)\) via of hat functions. Using properties of the generalized hat basis functions and fractional Brownian motion, new stochastic operational matrix of integration is achieved and the nonlinear stochastic equation is transformed into nonlinear system of algebraic equations which by solving it, an approximation solution with high accuracy is obtained. In addition, error analysis of the method is investigated, and by some examples, efficiency and accuracy of the suggested method are shown.     

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.     

Solutions to BSDEs driven by both standard and fractional Brownian motions

The backward stochastic differential equations driven by both standard and fractional Brownian motions (or, in short, SFBSDE) are studied. A Wick-Itô stochastic integral for a fractional Brownian motion is adopted. The fractional Itô formula for the standard and fractional Brownian motions is provided. Introducing the concept of the quasi-conditional expectation, we study some its properties. Using the quasi-conditional expectation, we also discuss the existence and uniqueness of solutions to general SFBSDEs, where a fixed point principle is employed. Moreover, solutions to linear SFBSDEs are investigated. Finally, an explicit solution to a class of linear SFBSDEs is found.     

Integral representation of renormalized self-intersection local times

In this paper we apply Clark-Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the d-dimensional fractional Brownian motion with Hurst parameter H∈(0,1). As a consequence, we derive the existence of some exponential moments for this random variable.     

Stochastic heat equation driven by fractional noise and local time

The aim of this paper is to study the d-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of a fractional Brownian motion with Hurst parameter H ∈ (0,1) in time. Two types of equations are considered. First we consider the equation in the Itô-Skorohod sense, and later in the Stratonovich sense. An explicit chaos expansion for the solution is obtained. On the other hand, the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the Brownian motion.     

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.     

Cumulant operators and moments of the Itô and Skorohod integrals

We propose a formula for the computation of the moments of all orders of Itô and Skorohod stochastic integrals with respect to Brownian motion, based on cumulant operators defined by the Malliavin calculus. Some characterizations of Gaussian distributions for stochastic integrals are recovered as a consequence.     

Brownian motion in the framework of quaternion analysis

Abstract

We study Brownian motion in the setting of quaternion analysis. We give a quaternion version of the Itô’s integral.     

Mixed Stochastic Differential Equations: Existence and Uniqueness Result

In this paper we establish an existence and uniqueness result 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 under a weaker condition than the Lipschitz one.     

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.