A change of variable formula with Itô correction term

Details A change of variable formula with Itô correction term A change of variable formula with Itô correction term

2008-02-22

arxiv.org/abs/0802.3356v4

Annals of Probability 2010, Vol. 38, No. 5, 1817-1869

Mathematics

Probability

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

Scientific paper

10.1214/09-AOP523

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 $F$.

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