Variations of the solution to a stochastic heat equation

Swanson, Jason

Ann. Probab., Tome 35 (2007) no. 1, p. 2122-2159 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2007-11-14
Classification:  Quartic variation,  quadratic variation,  stochastic partial differential equations,  stochastic integration,  long-range dependence,  iterated Brownian motion,  fractional Brownian motion,  self-similar processes,  60F17,  60G15,  60G18,  60H05,  60H15