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

TL;DR

This paper proves the uniqueness in law of stochastic convolution processes driven by Lévy noise, showing that identical laws of the driving processes imply identical laws of the convolution processes in a Banach space setting.

Contribution

It establishes a law uniqueness result for stochastic convolutions driven by Lévy noise, extending the understanding of their probabilistic behavior in Banach spaces.

Findings

Law of the stochastic convolution process is uniquely determined by the law of the driving Lévy noise.

Equivalent laws of the input processes imply equivalent laws of the convolution processes.

The result applies to processes in Banach spaces with specific regularity and measurability conditions.

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 $L^p([0,T],{L}^{p}(Z,\nu ;E))\times \CM_I([0,T]\times Z)$ 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,\nu ;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.