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

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

2010-10-28

arxiv.org/abs/1010.5941v1

Mathematics

Probability

Scientific paper

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 processes such that the laws on $L^p([0,T],E)\times \CM_I([0,T]\times S)$ 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 $L^p([0,T],E)\times \CM_I([0,T]\times S)\times\DD([0,T];X)\cap L^p([0,T];B )$ of the triples $(\xi_i,\eta_i,u_i)$, $i=1,2$, are equal as well. Here, by $\DD([0,T];X)$ we denote the Skorohod space of $X$-valued processes.

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