Uniqueness in Law of the stochastic convolution process driven by Lévy noise

Abstract

We will give a proof of the following fact. If $\mathfrak{A}_1$ and $\mathfrak{A}_2$, $\tilde \eta_1$ and $\tilde \eta_2$, $\xi_1$ and $\xi_2$ are two examples of filtered probability spaces, time homogeneous compensated Poisson random measures, and progressively measurable Banach space valued processes such that the laws on $L^p([0,T],{L}^{p}(Z,\nu ;E))\times \mathcal{M}_I([0,T]\times Z)$ of the pairs $(\xi_1,\eta_1)$ and $(\xi_2,\eta_2)$, are equal, and $u_1$ and $u_2$ are the corresponding stochastic convolution processes, then the laws on $ (\mathbb{D}([0,T];X)\cap L^p([0,T];B)) \times L^p([0,T],{L}^{p}(Z,\nu ;E))\times \mathcal{M}_I([0,T]\times Z) $, where $B \subset E \subset X$, of the triples $(u_i,\xi_i,\eta_i)$, $i=1,2$, are equal as well. By $\mathbb{D}([0,T];X)$ we denote the Skorokhod space of $X$-valued processes.