Stochastic PDEs in $\mathcal{S}^\prime$ for SDEs driven by L\'evy noise

TL;DR

This paper establishes a framework connecting finite-dimensional SDEs driven by Lévy processes to stochastic PDEs in the space of tempered distributions, proving existence, uniqueness, and translation invariance of solutions.

Contribution

It introduces a novel formulation of Lévy-driven SDEs as stochastic PDEs in the space of tempered distributions, extending previous diffusion results.

Findings

Proves existence and uniqueness of solutions to the stochastic PDEs

Constructs solutions with translation invariance property

Extends diffusion process results to Lévy processes

Abstract

In this article we show that a finite dimensional stochastic differential equation driven by a L\'evy process can be formulated as a stochastic partial differential equation. We prove the existence and uniqueness of strong solutions of such stochastic PDEs. The solutions that we construct have the `translation invariance' property. The special case of this correspondence for diffusion processes was proved in [Rajeev, Translation invariant diffusion in the space of tempered distributions, Indian J. Pure Appl. Math. 44 (2013), no.~2, 231--258].