Stochastic Convolutions Driven by Martingales: Maximal Inequalities and Exponential Integrability

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 Zygmund's 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. Keywords: Exponential tail estimatesMaximal inequalityStochastic convolutionsStochastic partial differential equationsUnitary dilationsMathematics Subject Classification: 60H15 Acknowledgments This research has been supported in part by the Grant APART 700 of the Austrian Academy of Science, the Austrian Science Foundation Grant P17273-N12, the MŠMT ČR/ÖAD Grant KONTAKT 2001-5, and by the GAČR Grants no. 201/01/1197 and 201/04/0750. We are indebted to A. S. Cherny who familiarized us with an example of a two dimensional martingale violating (0.4). Thanks are due to Z. Brzeźniak, S. Peszat, and J. Zabczyk for very useful discussions concerning the article.