Authors

K. Burdzy;J. Swanson

Comments

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Abbreviated Journal Title

Ann. Probab.

Keywords

Stochastic integration; quartic variation; quadratic variation; stochastic partial differential equations; long-range dependence; iterated Brownian motion; fractional Brownian motion; self-similar; processes; FRACTIONAL BROWNIAN-MOTION; STOCHASTIC INTEGRALS; LIMIT-THEOREMS; HURST; INDEX; Statistics & Probability

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 Ito sense. We show that for sufficiently differentiable functions g(x, t), a stochastic integral 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 Ito integral with respect to a Brownian motion that is independent of F.

Journal Title

Annals of Probability

Volume

38

Issue/Number

5

Publication Date

1-1-2010

Document Type

Article

Language

English

First Page

1817

Last Page

1869

WOS Identifier

WOS:000281425000003

ISSN

0091-1798

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