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 Fr\'{e}chet 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 $(E, \| \cdot\|)$.
We consider the following stochastic
convolution process
\begin{align*}
u(t)=\int_0^t\int_ZS(t-s)\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 c\`{a}dl\`{a}g 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 $C=C(q,p)$.
| Original language | English |
|---|---|
| Pages (from-to) | 937-956 |
| Number of pages | 20 |
| Journal | Annales Henri Poincare |
| Volume | 53 |
| Publication status | Published - May 2017 |
Bibliographical note
This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details.Keywords
- Stochastic convolution
- martingale type $p$ Banach space,
- Poisson random measure