## Maximal inequality of Stochastic convolution driven by Poisson random measure on Banach spaces

Research output: Working paper

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Date Published - May 2010 24 Undefined/Unknown

### Abstract

Assume that $E$ is an $M$-type $p$ Banach space with $q$-th, $q\geq p$, power of the norm is of $C^2$-class. We consider the stochastic convolution
$$u(t)=\int_0^t\int_ZS(t-s)\xi(s,z)\tilde{N}(ds,dz),$$ where $S$ is a $C_0$-semigroup of contractions on $E$ and $\tilde{N}$ is a compensated Poisson random measure. We derive a maximal inequality for a c\{a}dl\{a}g modification $\tilde{u}$ of $u$, $$\E\sup_{0\leq s\leq t}|\tilde{u}(s)|_E^{q'}\leq C\ \E\left(\int_0^t\int_Z|\xi(s,z)|_E^{p}N(ds,dz)\right)^{\frac{q'}{p}},$$ for every $0<q'< \infty$ and some constant $C>0$.

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