On It\^o formulas for jump processes

TL;DR

This paper revisits and revises the classical Itô formula for jump processes, extending it to infinite-dimensional $L_p$-valued jump processes relevant for stochastic PDEs.

Contribution

It introduces a generalized Itô formula for jump processes in infinite-dimensional spaces, expanding the applicability to stochastic PDEs.

Findings

Revised Itô formula for finite-dimensional jump processes

Generalization to $L_p$-valued jump processes

Applications in stochastic PDE theory

Abstract

A well-known It\^o formula for finite dimensional processes, given in terms of stochastic integrals with respect to Wiener processes and Poisson random measures, is revisited and is revised. The revised formula, which corresponds to the classical It\^o formula for semimartingales with jumps, is then used to obtain a generalisation of an important infinite dimensional It\^o formula for continuous semimartingales proved by Krylov to a class of $L_p$-valued jump processes. This generalisation is motivated by applications in the theory of stochastic PDEs.