Maximal inequality of Stochastic convolution driven by compensated Poisson random measures in Banach spaces
Abstract
Let $(E, \ \cdot\)$ be a Banach space such that, for some $q\geq 2$, the function $x\mapsto \x\^q$ is of $C^2$ class and its first and second Fréchet derivatives are bounded by some constant multiples of $(q1)$th power of the norm and $(q2)$th power of the norm and let $S$ be a $C_0$semigroup of contraction type on $(E, \ \cdot\)$. We consider the following stochastic convolution process \begin{align*} u(t)=\int_0^t\int_ZS(ts)\xi(s,z)\,\tilde{N}(\mathrm{d} s,\mathrm{d} z), \;\;\; t\geq 0, \end{align*} where $\tilde{N}$ is a compensated Poisson random measure on a measurable space $(Z,\mathcal{Z})$ and $\xi:[0,\infty)\times\Omega\times Z\rightarrow E$ is an $\mathbb{F}\otimes \mathcal{Z}$predictable function. We prove that there exists a càdlàg modification a $\tilde{u}$ of the process $u$ which satisfies the following maximal inequality \begin{align*} \mathbb{E} \sup_{0\leq s\leq t} \\tilde{u}(s)\^{q^\prime}\leq C\ \mathbb{E} \left(\int_0^t\int_Z \\xi(s,z) \^{p}\,N(\mathrm{d} s,\mathrm{d} z)\right)^{\frac{q^\prime}{p}}, \end{align*} for all $ q^\prime \geq q$ and $1<p\leq 2$ with $C=C(q,p)$.
 Publication:

arXiv eprints
 Pub Date:
 May 2010
 arXiv:
 arXiv:1005.1600
 Bibcode:
 2010arXiv1005.1600Z
 Keywords:

 Mathematics  Probability;
 60H15;
 60F10;
 60H05;
 60G57;
 60J75
 EPrint:
 This version is only very slightly updated as compared to the one from September 2015