Variations of the solution to a stochastic heat equation

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 It\^{o} 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.Comment: Published in at http://dx.doi.org/10.1214/009117907000000196 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Volatility estimation for stochastic PDEs using high-frequency observations

“…In particular, they have infinite quadratic variation and a nontrivial quartic variation, cf. Swanson (2007). Also, we find non-negligible negative autocovariances of increments such that the semi-martingale theory and martingale central limit theorems are not applicable to the marginal processes.…”

Temporal variation for fractional heat equations with additive white noise

“…Author details 1 Department of Statistics, Anhui Normal University, 1 East Beijing Rd., Wuhu, 241000, P.R. China.…”

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“…It is certainly possible to generalize to other types of Riemann sums. The 'Midpoint' sum, ⌊ can be shown to converge in probability for fBm with H > 1/4 (see [11]). The end point case H = 1/4 was considered in papers by Burdzy and Swanson [1], and Nourdin and Réveillac [7].…”