Stability in quadratic variation, with applications

TL;DR

This paper extends the stability properties of non-continuous Dirichlet processes under locally Lipschitz transformations, providing an Itô formula and applications to jump removal and integrator stability.

Contribution

It introduces a broader class of transformations under which non-continuous Dirichlet processes are stable, along with an Itô formula and practical applications.

Findings

Non-continuous Dirichlet processes are closed under certain Lipschitz maps.

An Itô formula for these transformations is established.

Applications include jump removal and stability of integrators.

Abstract

We show that non continuous Dirichlet processes, defined as in \cite{NonCont} are closed under a wide family of locally Lipschitz continuous maps (similar to the time-homogeneous variants of the maps considered in \cite{Low}) thus extending Theorem 2.1. from that paper. We provide an It\^o formula for these transforms and apply it to study of how $[f(X^n)-f(X)]\to 0$ when $X^n\to X$ (in some appropriate sense) for certain Dirichlet processes $\{X^n\}_n$, $X$ and certain locally Lipschitz continuous maps. We also consider how $[f_n(X^n)-f(X)]\to 0$ for $C^1$ maps $\{f_n\}_n$, $f$ when $f_n'\to f'$ uniformly on compacts. For applications we give examples of jump removal and stability of integrators.