A change of variable formula with 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 Itô 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 Itô integral with respect to a Brownian motion that is independent of $F$.

Published in at http://dx.doi.org/10.1214/09-AOP523 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)