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September 2010 A change of variable formula with Itô correction term
Krzysztof Burdzy, Jason Swanson
Ann. Probab. 38(5): 1817-1869 (September 2010). DOI: 10.1214/09-AOP523

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) 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.

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Krzysztof Burdzy. Jason Swanson. "A change of variable formula with Itô correction term." Ann. Probab. 38 (5) 1817 - 1869, September 2010. https://doi.org/10.1214/09-AOP523

Information

Published: September 2010
First available in Project Euclid: 17 August 2010

zbMATH: 1204.60044
MathSciNet: MR2722787
Digital Object Identifier: 10.1214/09-AOP523

Subjects:
Primary: 60H05
Secondary: 60G15 , 60G18 , 60H15

Keywords: fractional Brownian motion , iterated Brownian motion , long-range dependence , Quadratic Variation , Quartic variation , Self-similar processes , stochastic integration , Stochastic partial differential equations

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 5 • September 2010
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