Uniqueness in Law of the stochastic convolution process driven by L\'evy noise

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 $L^p([0,T],{L}^{p}(Z,\nu ;E))\times \CM_I([0,T]\times Z)$ofthepairs$(\xi_1,\eta_1)$and$(\xi_2,\eta_2)$$i=1,2$,areequal,and$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,\nu ;E))\times \CM_I([0,T]\times Z) $,where$B \subset E \subset X$,ofthetriples$(u_i,\xi_i,\eta_i)$,$i=1,2$, are equal as well. By $\text{DD}([0,T];X)$ we denote the Skorokhod space of $X$-valued processes.