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.
@article{1282053773,
author = {Burdzy, Krzysztof and Swanson, Jason},
title = {A change of variable formula with It\^o correction term},
journal = {Ann. Probab.},
volume = {38},
number = {1},
year = {2010},
pages = { 1817-1869},
language = {en},
url = {http://dml.mathdoc.fr/item/1282053773}
}
Burdzy, Krzysztof; Swanson, Jason. A change of variable formula with Itô correction term. Ann. Probab., Tome 38 (2010) no. 1, pp. 1817-1869. http://gdmltest.u-ga.fr/item/1282053773/