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 $L^p([0,T],{L}^{p}(Z,ν;E))\times \CM_I([0,T]\times Z)$ of the pairs $({\xi }_1,{\eta }_1)$ and $({\xi }_2,{\eta }_2)$ %, $i=1,2$, are equal, and $u_1$ and $u_2$ are the corresponding stochastic convolution processes, then the laws on $ (\DD([0,T];X)\cap L^p([0,T];B)) \times L^p([0,T],{L}^{p}(Z,ν;E))\times \CM_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 $\DD([0,T];X)$ we denote the Skorokhod space of $X$-valued processes.