Maximal inequality of Stochastic convolution driven by compensated Poisson random measures in Banach spaces

Details Maximal inequality of Stochastic convolution driven by compensated Poisson random measures in Banach spaces Maximal inequality of Stochastic convolution driven by compensated Poisson random measures in Banach spaces

2010-05-10

arxiv.org/abs/1005.1600v1

Mathematics

Probability

24pages

Scientific paper

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 $00$.

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