The Annals of Probability

A change of variable formula with Itô correction term

Krzysztof Burdzy and Jason Swanson

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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) 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.

Article information

Ann. Probab. Volume 38, Number 5 (2010), 1817-1869.

First available in Project Euclid: 17 August 2010

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Mathematical Reviews number (MathSciNet)

Primary: 60H05: Stochastic integrals
Secondary: 60G15: Gaussian processes 60G18: Self-similar processes 60H15: Stochastic partial differential equations [See also 35R60]

Stochastic integration quartic variation quadratic variation stochastic partial differential equations long-range dependence iterated Brownian motion fractional Brownian motion self-similar processes


Burdzy, Krzysztof; Swanson, Jason. A change of variable formula with Itô correction term. Ann. Probab. 38 (2010), no. 5, 1817--1869. doi:10.1214/09-AOP523.

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