Preprint2012

Variations of the solution to a stochastic heat equation

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Stochastic processes and financial applications Stochastic processes and statistical mechanics Mathematical and Theoretical Analysis

Abstract

We consider the solution to a stochastic heat equation. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time, F(t), has a nontrivial quartic variation. This process, therefore, has infinite quadratic variation and is not a semimartingale. It follows that the classical Itô calculus does not apply. Motivated by heuristic ideas about a possible new calculus for this process, we are led to study modifications of the quadratic variation. Namely, we modify each term in the sum of the squares of the increments so that it has mean zero. We then show that these sums, as functions of t, converge weakly to Brownian motion. 1. Introduction. Consider

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References

1968Fractional Brownian Motions, Fractional Noises and Applications 2007Brownian Motion and Stochastic Calculus 2006The Malliavin Calculus and Related Topics 1998Differential equations driven by rough signals 2005Central limit theorems for sequences of multiple stochastic integrals 1998Asymptotic error distributions for the Euler method for stochastic differential equations 1993Forward, backward and symmetric stochastic integration 2002Stochastic analysis, rough path analysis and fractional Brownian motions

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