Fractional geometric mean-reversion processes

We use a method developed in Carmona et al. (2003) [2] to study the fractional geometric mean-reversion processes. Our obtained results hold for any .     

Various types of stochastic integrals with respect to fractional Brownian sheet and their applications

In this paper, we develop a stochastic calculus related to a fractional Brownian sheet as in the case of the standard Brownian sheet. Let be a fractional Brownian sheet with Hurst parameters H=(H₁,H₂), and (²[0,1],B(²[0,1]),μ) a measure space. By using the techniques of stochastic calculus of variations, we introduce stochastic line integrals along all sufficiently smooth curves γ in ²[0,1], and four types of stochastic surface integrals: , i=1,2, , , , . As an application of these stochastic integrals, we prove an Itô formula for fractional Brownian sheet with Hurst parameters H₁,H₂∈(1/4,1). Our proof is based on the repeated applications of Itô formula for one-parameter Gaussian process.     

Fractional Bilinear Stochastic Equations with the Drift in the First Fractional Chaos

Abstract

In the paper we compute the explicit form of the fractional chaos decomposition of the solution of a fractional stochastic bilinear equation with the drift in the fractional chaos of order one and initial condition in a finite fractional chaos. The large deviations principle is also obtained for the one-dimensional distributions of the solution of the equation perturbed by a small noise.     

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.     

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.     

A stochastic maximum principle for processes driven by fractional Brownian motion

We prove a stochastic maximum principle for controlled processes X(t)=X^(u)(t) of the form

dX(t)=b(t,X(t),u(t)) dt+σ(t,X(t),u(t)) dB^(H)(t),

where B^(H)(t) is m-dimensional fractional Brownian motion with Hurst parameter . As an application we solve a problem about minimal variance hedging in an incomplete market driven by fractional Brownian motion.     

Brownian and fractional Brownian stochastic currents via Malliavin calculus

By using Malliavin calculus and multiple Wiener-Itô integrals, we study the existence and the regularity of stochastic currents defined as Skorohod (divergence) integrals with respect to the Brownian motion and to the fractional Brownian motion. We consider also the multidimensional multiparameter case and we compare the regularity of the current as a distribution in negative Sobolev spaces with its regularity in the Watanabe spaces.     

Asymptotic behavior of linear fractional stochastic differential equations with time-varying delays

In this paper we give a sufficient condition for the exponential asymptotic behavior of solutions of a general class of linear fractional stochastic differential equations with time-varying delays. Our obtained results allow us to employ the theories developed for the deterministic systems and to illustrate this, some examples are provided.     

Stochastic evolution equations with fractional Brownian motion

In this paper linear stochastic evolution equations driven by infinite-dimensional fractional Brownian motion are studied. A necessary and sufficient condition for the existence and uniqueness of the solution is established and the spatial regularity of the solution is analyzed; separate proofs are required for the cases of Hurst parameter above and below 1/2. The particular case of the Laplacian on the circle is discussed in detail. Mathematics Subject Classification (2000): 60H15, 60G15     

Extensions of the well-poised and elliptic well-poised Bailey lemma

We establish a number of extensions of the well-poised Bailey lemma and elliptic well-poised Bailey lemma. As application we prove some new transformation formulae for basic and elliptic hyper-geometric series, and embed some recent identities of Andrews, Berkovich and Spiridonov in a well-poised Bailey tree.     

Operators associated with a stochastic differential equation driven by fractional Brownian motions

In this paper, by using a Taylor type development, we show how it is possible to associate differential operators with stochastic differential equations driven by fractional Brownian motions. As an application, we deduce that invariant measures for such SDE’s must satisfy an infinite dimensional system of partial differential equations.     

Approximation of the finite dimensional distributions of multiple fractional integrals

We construct a family Inεt(f) of continuous stochastic processes that converges in the sense of finite dimensional distributions to a multiple Wiener-Itô integral with respect to the fractional Brownian motion. We assume that and we prove our approximation result for the integrands f in a rather general class.     

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.     

On Fractional Brownian Processes

We use Liouville spaces in order to prove the existence of some different fractional -Brownian motion ( 0 < 1 ), or fractional ( , )-Brownian sheets. There are also applications to the Wiener stochastic integral with respect to these -Brownian.     

Local times of stochastic differential equations driven by fractional Brownian motions

In this paper, we study the existence and (Hölder) regularity of local times of stochastic differential equations driven by fractional Brownian motions. In particular, we show that in one dimension and in the rough case $H<1/2$, the Hölder exponent (in $t$) of the local time is $1?H$, where $H$ is the Hurst parameter of the driving fractional Brownian motion.     

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.     

Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion

We prove the Malliavin regularity of the solution of a stochastic differential equation driven by a fractional Brownian motion of Hurst parameter H>0.5$H>0.5$. The result is based on the Fréchet differentiability with respect to the input function for deterministic differential equations driven by Hölder continuous functions. It is also shown that the law of the solution has a density with respect to the Lebesgue measure, under a suitable nondegeneracy condition.