Temporal quadratic and higher order variation for the nonlinear stochastic heat equation and applications to parameter estimation

Abstract

We consider the stochastic heat equation which includes a fractional power of the Laplacian of order $\alpha \in (1, 2]$ and it is driven by a nonlinear space-time Gaussian white noise. We study two types of power variations for the solution to this equation: the renormalized quadratic variation and the power variation of order $\frac{2\alpha}{\alpha -1}$, both over an equidistant partition of the unit interval. We prove that these two sequences admit nontrivial limits when the mesh of the partition goes to zero. We apply these results to identify certain parameters of the stochastic heat equation.