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

Abstract

Let $(E,\Vert \cdot \Vert )$ be a Banach space such that, for some $q\ge 2$, the function $x\mapsto \Vert x{\Vert }^q$ is of $C^2$ class and its first and second Fréchet derivatives are bounded by some constant multiples of $(q-1)$-th power of the norm and $(q-2)$-th power of the norm and let $S$ be a $C_0$-semigroup of contraction type on $(E,\Vert \cdot \Vert )$. We consider the following stochastic convolution process \begin{align*} u(t)=\int_0^t\int_ZS(t-s)ξ(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 $\xi :[0,\infty )\times \Omega \times Z\to E$ 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 \|ξ(s,z) \|^{p}\,N(\mathrm{d} s,\mathrm{d} z)\right)^{\frac{q^\prime}{p}}, \end{align*} for all $q^{\prime }\ge q$ and $1<p\le 2$ with $C=C(q,p)$.

This version is only very slightly updated as compared to the one from September 2015

Cited by in corpus (4)

• Stochastic Reaction-diffusion Equations Driven by Jump Processes
• On maximal inequalities for purely discontinuous martingales in infinite dimensions
• On the Exponential Stability of Switching-Diffusion Processes with Jumps
• A maximal inequality for stochastic convolutions in 2-smooth Banach spaces