Stability in quadratic variation

TL;DR

This paper investigates the stability of quadratic variations in sequences of cadlag processes, establishing conditions under which convergence of processes implies convergence of their quadratic variations, with implications for stochastic integration.

Contribution

It provides new stability results for quadratic variations in cadlag processes, including a novel Itô decomposition for strong sense quadratic variations.

Findings

Stability of quadratic variations when $f$ is in $C^1$ and processes are Dirichlet.

Relaxed conditions for strong sense quadratic variations with independent parts of processes.

Introduction of a new Itô decomposition for strong sense quadratic variations.

Abstract

Consider a sequence of cadlag processes $\{X^n\}_n$, and some fixed function $f$. If $f$ is continuous then under several modes of convergence $X^n\to X$ implies corresponding convergence of $f(X^n)\to f(X)$, due to continuous mapping. We study conditions (on $f$, $\{X^n\}_n$ and $X$) under which convergence of $X^n\to X$ implies $\left[f(X^n)-f(X)\right]\to 0$. While interesting in its own right, this also directly relates (through integration by parts and the Kunita-Watanabe inequality) to convergence of integrators in the sense $\int_0^t Y_{s-}df(X^n_s)\to\int_0^t Y_{s-}df(X_s)$. We use two different types of quadratic variations, weak sense and strong sense which our two main results deal with. For weak sense quadratic variations we show stability when $f\in C^1$, $\{X^n\}_n,X$ are Dirichlet processes defined as in \cite{NonCont} $X^n\xrightarrow{a.s.}X$, $[X^n-X]\xrightarrow{a.s.}0$ and $\{(X^n)^*_t\}_n$ is bounded in probability. For strong sense quadratic variations we are able to relax the conditions on $f$ to being the primitive function of a cadlag function but with the additional assumption on $X$, that the continuous and discontinuous parts of $X$ are independent stochastic processes (this assumption is not imposed on $\{X^n\}_n$ however), and $\{X^n\}_n,X$ are Dirichlet processes with quadratic variations along any stopping time refining sequence. To prove the result regarding strong sense quadratic variation we prove a new It\^o decomposition for this setting.