Jan Seidler

Stochastic Convolutions Driven by Martingales: Maximal Inequalities and Exponential Integrability

Abstract Stochastic convolutions driven by a local martingale M in a Hilbert space are studied in the case when S(t) is a strongly continuous semigroup of contractions. Very simple proofs of the maximal inequality and exponential tail estimates are given by using unitary dilations and Zygmunds extrapolation theorem. Applications to stochastic convolutions driven by Poisson random measures are provided. A part of the results is then generalized to stochastic convolutions in L q -spaces.

On Weak Solutions of Stochastic Differential Equations

A new proof of existence of weak solutions to stochastic differential equations with continuous coefficients based on ideas from infinite-dimensional stochastic analysis is presented. The proof is fairly elementary, in particular, neither theorems on representation of martingales by stochastic integrals nor results on almost sure representation for tight sequences of random variables are needed.

Weak convergence of infinite-dimensional diffusions 1

Continuous dependence - in the sense of weak convergence of laws - of martingale solutions to stochastic partial differential equations on coefficients is studied, the results obtained being applicable to equations with rapidly oscillating coefficients. In the proofs, Gatareks and Goldys’ recent approach to martingale solutions is substantially used

On Weak Solutions of Stochastic Differential Equations II

In the first part of this article a new method of proving existence of weak solutions to stochastic differential equations with continuous coefficients having at most linear growth was developed. In this second part, we show that the same method may be used even if the linear growth hypothesis is replaced with a suitable Lyapunov condition.

A note on continuous-time stochastic approximation in infinite dimensions

We find sufficient conditions for convergence of a continuous-time Robbins-Monro type stochastic approximation procedure in infinite dimensional Hilbert spaces in terms of Lyapunova functions, the variational approach to stochastic partial differential equations being used as the main tool.