Parameter Estimation for Discretely Sampled SPDEs

We consider a parameter estimation problem for one dimensional stochastic heat equations, when data is sampled discretely in time or spatial component. We establish some general results on derivation of consistent and asymptotically normal estimators based on computation of the \(p\)-variations of stochastic processes and their smooth perturbations. We apply these results to the considered SPDEs, by using some convenient representations of the solutions. For some equations such results were ready available, while for other classes of SPDEs we derived the needed representations along with their statistical asymptotical properties. We prove that the real valued parameter next to the Laplacian, and the constant parameter in front of the noise (the volatility) can be consistently estimated by observing the solution at a fixed time and on a discrete spatial grid, or at a fixed space point and at discrete time instances of a finite interval, assuming that the mesh-size goes to zero.

Event Topic

Mathematical Finance, Stochastic Analysis, and Machine Learning