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

TY - UNPB

T1 - Maximal inequality of Stochastic convolution driven by Poisson random measure on Banach spaces

AU - Brzezniak, Zdzislaw

AU - Hausenblas, Erika

AU - Zhu, Jiahui

PY - 2010/5

Y1 - 2010/5

N2 - 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 $00$.

AB - 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 $00$.

M3 - Preprint

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

ER -