Stochastic Analysis

Friday January 22

Frederic Utzet (Universitat Autònoma de Barcelona)
Stein's method and Malliavin calculus of Poisson functionals

Joint work with Giovanni Peccati, Université Paris Ouest - Nanterre; Murad Taqqu, Boston University; Josep Lluís Solé, Universitat Autònoma de Barcelona

In a recent series of papers by Nourdin, Peccati, Réveillac and Reiner it has been shown that Stein's method can be effectively combined with Malliavin calculus on a Gaussian space in order to obtain explicit bounds for the normal and non-normal approximation of smooth functionals of Gaussian fields, mainly multiple Wiener-Itô integrals.

In this talk we will present how these results can be extended to the approximation (in the Wasserstein distance) of functionals of Poisson measures. In particular we apply these ideas to a sequence of multiple Wiener-Itô integrals with respect to a Poisson measure.

As in the Gaussian case, the main ingredients of the analysis are the following:

1. A set of Stein differential equations, relating the normal approximation in the Wasserstein distance to first order differential operators.

2. A derivative operator D, acting on square-integrable random variables.

3. An integration by parts formula, involving the adjoint operator of D, the Skorohod integral.

4. A pathwise representation of D which, in the Poisson case, involves a difference operator.

Josep Lluís Solé i Clivillés (Universitat Autònoma de Barcelona)
Malliavin Calculus for Lévy Processes via Derivative Operators

We develop a Malliavin calculus for Lévy processes based on derivative operators. This approach include the classical Malliavin calculus for Gaussian processes and the one for the Poisson process introduced by Carlen and Pardoux. As an application we analyze sufficient conditions for the absolute continuity of the law of some stochastic differential equations driven by Lévy processes.

Jana Šnupárková (Charles University, Prague)
Stochastic bilinear differential equation with fractional noise in infinite dimension

We will study the infinite-dimensional stochastic bilinear equation with one-dimension fractional noise in the singular case H<1/2.

Erika Hausenblas (University of Salzburg)
SPDEs of reaction-diffusion type driven by Levy noise

The topic of the talk are Spdes driven by Poisson random measure of reaction diffusion type. In the first part I will shortly outline Poisson random measures, then I will dwell shortly on the connection between Levy processes and Poisson random measures. Moreover, I will introduce space time Levy noise respective space time Poissonian noise. Then, the stochastic integral with respect to Poisson random measures in Banach spaces will be treated and some inequalities pointed out.

In the second part of the talk Stochastic partial Differential Equations driven by Possion random measures will be considered. Here, I want to present some result concerning SPDEs driven by only continuous coefficients. In this case, one can show only the existstence of martingale solutions.

The topic of the talk rests on the following papers:
Brzezniak and Hausenblas, Maximale inequality of the stochastic convolution process, PTRF, 2008.
Brzezniak and Hausenblas, Uniqueness of the Ito integral driven by Poisson random measure, to appear in the Ascona proceedings, 2009.
Brzezniak and Hausenblas, Uniqueness of the stochastic convolution process driven by Poisson random measure, submitted, 2009.
Brzezniak and Hausenblas, SPDEs of reaction diffusion type Poisson random measure driven by Poisson random measure, submitted, 2009.

Jan Seidler (Academy of Sciences, Prague)
Moment inequalities for stochastic integrals in 2-smooth Banach spaces

Martin Ondreját (Academy of Sciences, Prague)
Stochastic wave equations

A survey of basic and recent results in the theory of wave equations driven by Wiener processes.

Carles Rovira Escofet (Universitat de Barcelona)
Delay differential equations with hereditary drift driven by fractional Brownian motion with Hurst parameter H > 1/2

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. We show that when the delay goes to zero the solutions to these equations converge, almost surely and in Lp, to the solution for the equation without delay.

We consider also the case of stochastic differential delay equations with reflection.

Xavier Bardina (Universitat Autònoma de Barcelona)
Relationships between fractional, bifractional and subfractional Brownian motions

Jan Pospíšil ( University of West Bohemia, Pilsen )
Statistical inference for SPDEs driven by fractional noise

We will discuss a parameter estimation problem for a diagonalizable stochastic evolution equation driven by additive noise that is white in space and fractional in time. Some asymptotic properties of the maximum likelihood estimator, as the number of the Fourier coefficients of the solution increases, will be investigated. Necessary and sufficient conditions for consistency and asymptotic normality will be presented.

Jan Bártek ( Charles University, Prague )
Stochastic porous medium equation with fractional Brownian motion

In this talk we study the relation between solutions to a certain type of nonlinear partial differential equation and solutions to its stochastic analogy driven by a fractional Brownian motion with Hurst index H > 1/2. We obtain a formula for the solutions which is used to study properties of the solution to the stochastic porous medium equation.

Josef Janák (Charles University, Prague)
Ornstein-Uhlenbeck bridge

Fractional Ornstein-Uhlenbeck processes in finite-dimensional spaces will be studied. First, known results on existence and uniqueness on solutions to linear stochastic differential equations, which define Ornstein-Uhlenbeck processes, will be recalled. Then we will introduce general Gaussian bridges and their representations. Finally, we will derive a formula for a Gaussian bridge driven by an Ornstein-Uhlenbeck process.

Jana Lipková (Charles University, Prague)
Reversible Reactions in Reaction-Diffusion Stochastic Simulation Algorithms

Introduction to some mathematical models used in modelling of chemical reactions. Drawback of these models when reversible reactions are considered. Introduction to Andrew and Bray modelling of reversible reactions and related problems. Introduction and closer look on stochastic approach with implementation of probability.

Bohdan Maslowski (Charles University, Prague)
LQ control for FBM-driven SPDEs

The linear-quadratic control problem is studied for stochastic equation in infinite dimensions that is driven by the fractional Brownian motion. The existence and uniqueness of the optimal control is proved and, under more restrictive assumptions, the optimal conrtol is given in the feedback form. The results are applicable to parabolic and hyperbolic stochastic PDEs with distributed noise and to parabolic systems with boundary/pointwise noise and control.