David Nualart

Stochastic Calculus with Anticipating Integrands

SummaryWe study the stochastic integral defined by Skorohod in [24] of a possibly anticipating integrand, as a function of its upper limit, and establish an extended Itô formula. We also introduce an extension of Stratonovichs integral, and establish the associated chain rule. In all the results, the adaptedness of the integrand is replaced by a certain smoothness requirement.

Central limit theorems for sequences of multiple stochastic integrals

We characterize the convergence in distribution to a standard normal law for a sequence of multiple stochastic integrals of a fixed order with variance converging to 1. Some applications are given, in particular to study the limiting behavior of quadratic functionals of Gaussian processes.

Chaotic and predictable representations for Levy processes

The only normal martingales which posses the chaotic representation property and the weaker predictable representation property and which are at the same time also Levy processes, are in essence Brownian motion and the compensated Poisson process. For a general Levy process (satisfying some moment conditions), we introduce the power jump processes and the related Teugels martingales. Furthermore, we orthogonalize the Teugels martingales and show how their orthogonalization is intrinsically related with classical orthogonal polynomials. We give a chaotic representation for every square integral random variable in terms of these orthogonalized Teugels martingales. The predictable representation with respect to the same set of orthogonalized martingales of square integrable random variables and of square integrable martingales is an easy consequence of the chaotic representation.

Stochastic integration with respect to the fractional Brownian motion

We develop a stochastic calculus for the fractional Brownian motion with Hurst parameter using the techniques of the Malliavin calculus. We establish estimates in L p , maximal inequalities and a continuity criterion for the stochastic integral. Finally, we derive an Itôs formula for integral processes.

Evolution equations driven by a fractional Brownian motion

Abstract In this paper we study nonlinear stochastic evolution equations in a Hilbert space driven by a cylindrical fractional Brownian motion with Hurst parameter H> 1 2 and nuclear covariance operator. We establish the existence and uniqueness of a mild solution under some regularity and boundedness conditions on the coefficients and for some values of the parameter H. This result is applied to stochastic parabolic equation perturbed by a fractional white noise. In this case, if the coefficients are Lipschitz continuous and bounded the existence and uniqueness of a solution holds if H> d 4 . The proofs of our results combine techniques of fractional calculus with semigroup estimates.

Generalized stochastic integrals and the malliavin calculus

SummaryThe paper first reviews the Skorohod generalized stochastic integral with respect to the Wiener process over some general parameter space T and its relation to the Malliavin calculus as the adjoint of the Malliavin derivative. Some new results are derived and it is shown that every sufficiently smooth process {ut, t∈T} can be decomposed into the sum of a Malliavin derivative of a Wiener functional, and a process whose generalized integral over T vanishes. Using the results on the generalized integral, the Bismut approach to the Malliavin calculus is generalized by allowing non adapted variations of the Wiener process yielding sufficient conditions for the existence of a density which is considerably weaker than the previously known conditions.Let ei be a non-random complete orthonormal system on T, the Ogawa integral ∫u

Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than

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White noise driven quasilinear SPDEs with reflection

W is defined as ∑ i (eiu) ∫ eidW where the integrals are Wiener integrals. Conditions are given for the existence of an intrinsic Ogawa integral i.e. independent of the choice of the orthonormal system and results on its relation to the Skorohod integral are derived.The transformation of measures induced by (W + ∫ u d μu non adapted is discussed and a Girsanov-type theorem under certain regularity conditions is derived.