A change of variable formula with It\^{o} correction term

TL;DR

This paper investigates a stochastic heat equation solution's non-semimartingale behavior, establishing a new change of variable formula with an Itô correction term for integrals involving the solution process.

Contribution

It introduces a novel change of variable formula for non-semimartingale processes derived from stochastic PDEs, including an Itô correction term involving an independent Brownian motion.

Findings

The process has nontrivial quartic variation.

A stochastic integral can be defined via discrete Riemann sums.

The change of variable formula includes an Itô correction term.

Abstract

We consider the solution $u(x,t)$ to a stochastic heat equation. For fixed $x$, the process $F(t)=u(x,t)$ has a nontrivial quartic variation. It follows that $F$ is not a semimartingale, so a stochastic integral with respect to $F$ cannot be defined in the classical It\^{o} sense. We show that for sufficiently differentiable functions $g(x,t)$, a stochastic integral $\int g(F(t),t)\,dF(t)$ exists as a limit of discrete, midpoint-style Riemann sums, where the limit is taken in distribution in the Skorokhod space of cadlag functions. Moreover, we show that this integral satisfies a change of variable formula with a correction term that is an ordinary It\^{o} integral with respect to a Brownian motion that is independent of $F$.