Skorohod integral of a product of two stochastic processes

In this note we provide sufficient conditions for the product of a predictable and a backward predictable process to be Skorohod integrable.Supported in part by NSF No. INT-9401109, NSA No. MDR-90494H2094, and ONR No. N-00014-96-1-0262.     

Fractional smoothness for the generalized local time of the indefinite Skorohod integral

Let be the indefinite Skorohod integral on Wiener space (Ω,H,P), and let Lt(x) be its the generalized local time introduced by Tudor in [C.A. Tudor, Martingale-type stochastic calculus for anticipating integral processes, Bernoulli 10 (2004) 313-325]. We prove that the generalized local time, as a nonlinear functional of ω, is in the fractional Sobolev spaces Dα,p ( and p>2) under some conditions imposed on the anticipating integrand u via the technique of Malliavin calculus and the K-method in the real interpolation theory. The result is optimal for the fractional Brownian motion with the Hurst parameter .     

Canonical Lévy process and Malliavin calculus

A suitable canonical Lévy process is constructed in order to study a Malliavin calculus based on a chaotic representation property of Lévy processes proved by Itô using multiple two-parameter integrals. In this setup, the two-parameter derivative _Dt,x$D_{t,x}$ is studied, depending on whether x=0$x=0$ or x≠0$x\ne 0$; in the first case, we prove a chain rule; in the second case, a formula by trajectories.     

The Skorohod integral in conuclear spaces

We define an anticipative stochastic integral with respect to a nonhomogeneous Wiener process in a dual of a nuclear space and investigate its basic properties. The theory is developed without the use of chaos expansions.     

Weak convergence of the Stratonovich integral with respect to a class of Gaussian processes

For a Gaussian process X$X$ and smooth function f$f$, we consider a Stratonovich integral of f(X)$f\left(X\right)$, defined as the weak limit, if it exists, of a sequence of Riemann sums. We give covariance conditions on X$X$ such that the sequence converges in law. This gives a change-of-variable formula in law with a correction term which is an Itô integral of f^?$f^{?}$ with respect to a Gaussian martingale independent of X$X$. The proof uses Malliavin calculus and a central limit theorem from Nourdin and Nualart (2010) [8]. This formula was known for fBm with H=1/6$H=1/6$ Nourdin et al. (2010) [9]. We extend this to a larger class of Gaussian processes.     

The Skorohod integral and the derivative operator of functional of a cylindrical Brownian motion

The paper presents a definition of the Skorohod integral of operator-valued processes and the derivative operator for functional of a cylindrical Brownian motionW on a Hilbert space. The method is based on the chaos expansions in terms of multiple Wiener integrals ofW.This research was partially supported by the U.S. Air Force Office of Scientific Research Contract No. F49620 85C 0144. The research of V. Pérez-Abreu was also supported by CONACYT Grant D111-904237.     

Wiener integrals, Malliavin calculus and covariance measure structure

We introduce the notion of covariance measure structure for square integrable stochastic processes. We define Wiener integral, we develop a suitable formalism for stochastic calculus of variations and we make Gaussian assumptions only when necessary. Our main examples are finite quadratic variation processes with stationary increments and the bifractional Brownian motion.     

A remark on the weak convergence of processes in the Skorohod topology

A Skorohod representation type theorem is proved for the weak convergence of stochastic processes in the Skorohod topology. This allows the time changes arising from the Skorohod topology to be considered as stochastic processes. While thenth time change processA _t ⁿ is not adapted to thenth filtration ( _t ⁿ ) _t0, it is possible to choose the processesA ⁿ such that they are adapted to, where, where _n is a sequence of constants decreasing to 0 asn tends to .Supported in part by NSF Grant No. DMS-9103454.     

Calcul des variations stochastique pour la mesure de densité uniforme

A gradient operator is defined for the functionals of a non-Markovian jump process Y whose jump times are given by uniform probability laws. The adjoint of this gradient extends the compensated stochastic integral with respect to Y. An explicit representation of the functionals of Y as stochastic integrals is obtained via a Clark formula in two different approaches. The associated Dirichlet forms is studied in order to obtain criteria for the existence and regularity of densities of random variables in infinite dimension.     

LINEAR SKOROHOD STOCHASTIC DIFFERENTIAL EQUATIONS IN THE TWO PARAMETER CASE

1.　IntroductionUsing Girsanov s transformation and Ito formula,Buckdahn (1 993) considered theequation with Skorohod integral:Xt=H ∫t0 b(Xs) ds ∫t0 a(Xs) d Ws,　 a.e.on A (1 .1 )where A is a bounded ball inΩ.For a linear Skorohod stochastic differential equation:Xt=H ∫t0 As Xsds ∫t0 Bs Xsd Ws,(1 ,2 )Shiota(1 986 ) constructed a solution by means of the Wiener- Itochaos decomposition,when(As) and(Bs) are deterministic processes and H is a random variable represented by…     

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 the Upper Bound of the Density for Truncated Stable Processes in Small Time

We present a point-wise concrete upper bounds in a small time for transition densities of truncated stable process in R ^d, which have singular Lévy measures. We provide several examples.     

The anticipative Stratonovich integral in conuclear spaces

We define an anticipative stochastic integral of Stratonovich type with respect to a nonhomogeneous Wiener process in the dual of a nuclear space and investigate its basic properties.This research was partially supported by Komitet Bada Naukowych, Grant 2 1094 91 01.     

A remark on the asymptotic expansion of density function of Wiener functionals

We consider the asymptotic expansion of density function of Wiener functionals as time tends to zero as in [S. Kusuoka, D.W. Stroock, Precise asymptotics of certain Wiener functionals, J. Funct. Anal. 99 (1991) 1-74], and give an explicit formula for the first coefficient.     

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 ).     

Anticipative Stochastic Differential Equations with Non-smooth Diffusion Coefficient

In this paper we prove the existence and uniqueness of the solutions to the one-dimensional linear stochastic differential equation with Skorohod integral Xt（ω）=η（w）＋∫^t 0 asXs（ω）dWs＋∫^t 0 bsXs（ω）ds, t∈[0,1] where （Ws） is the canonical Wiener process defined on the standard Wiener space （W,H,u）, a is non-smooth and adapted, but η and b may be anticipating to the filtration generated by （Ws）. The intention of the paper is to eliminate the regularity of the diffusion coefficient a in the Malliavin sense, in the existing literature. The idea is to approach the non-smooth diffusion coefficient a by smooth ones.