An It\=o formula in the space of tempered distributions

TL;DR

This paper extends the Itô formula to the space of tempered distributions, enabling new analysis of Lévy processes and solutions to stochastic differential equations within Hermite-Sobolev spaces.

Contribution

It introduces an Itô formula in the space of tempered distributions and applies it to Lévy processes for solving specific stochastic differential equations.

Findings

Extended Itô formula for semimartingales in tempered distributions

Established existence results for SDEs in Hermite-Sobolev spaces

Analyzed local time processes of such semimartingales

Abstract

We extend the It\=o formula \cite{MR1837298}*{Theorem 2.3} for semimartingales with rcll paths. We also comment on Local time process of such semimartingales. We apply the It\=o formula to L\'evy processes to obtain existence of solutions to certain classes of stochastic differential equations in the Hermite-Sobolev spaces.