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 ∫ g(F(t), t) d F(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.