Parametric family of SDEs driven by L\'evy noise

TL;DR

This paper investigates the existence and uniqueness of strong solutions for a family of stochastic differential equations driven by Lévy noise, which are linked to certain stochastic partial differential equations in the space of tempered distributions.

Contribution

It extends the theory of diffusion processes to a parametric family of Lévy-driven SDEs connected with stochastic PDEs in distribution spaces.

Findings

Proved existence and uniqueness of strong solutions for the class of SDEs.

Established the connection between these SDEs and stochastic PDEs in tempered distribution space.

Generalized previous diffusion process results to Lévy noise-driven equations.

Abstract

In this article we study the existence and uniqueness of strong solutions of a class of parameterized family of SDEs driven by L\'evy noise. These SDEs occurs in connection with a class of stochastic PDEs, which take values in the space of tempered distributions $\mathcal{S}^\prime$. 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].