A change of variable formula with Itô correction term

A change of variable formula with Itô correction term

Krzysztof Burdzy and Jason Swanson

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

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.

First Page: Show

Primary Subjects: 60H05

Secondary Subjects: 60G15, 60G18, 60H15

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

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Permanent link to this document: http://projecteuclid.org/euclid.aop/1282053773
Digital Object Identifier: doi:10.1214/09-AOP523
Mathematical Reviews number (MathSciNet): MR2722787

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