Variations of the solution to a stochastic heat equation

By Jason Swanson. 2005

71 citations recorded by the source corpus.

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 semimartin-gale. It follows that the classical Ito 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.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4ed79e68108099d1f5865855c5811df9fd002004

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Citations in the corpus

Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case H=1/4 (2008, background): “received a lot of attention, see [6, 12, 13, 14, 16] (see also the related works [17, 20, 21, 23 ]).”
Volatility estimation for stochastic PDEs using high-frequency observations (2017, background): “In particular, they have infinite quadratic variation and a nontrivial quartic variation, cf. Swanson (2007).”
Variations and estimators for selfsimilarity parameters via Malliavin calculus (2009, background): “, where X is a Gaussian process (fractional Brownian motion in [14, 16] and [17], and the solution of the heat equation driven by a space-time white noise in [26]) or the iterated Brownian motion in [18] and h is a regular deterministic function.”
Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion (2007, background): “Finally, let us mention that Swanson announced in [15] that, in a joint work with Burdzy, they will prove that the same also holds for the solution of the stochastic heat equation driven by a space–time white noise.”
Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes (2008, background): “Equivalently, B0 = 0 and E [ (Bs − Bt)2 ] = |t − s|2H .”