Parameter estimation problems for parabolic SPDEs

Igor Cialenco, Illinois Institute of Technology Weierstrass-Institute for Applied Analysis and Stochastics, Erhard-Schmidt-Hörsaal, Mohrenstraße 39, 10117 Berlin10:00 -12:30

In the first part of the talk we will discuss the parameter estimation problem using Bayesian approach for the drift coefficient of some linear (parabolic) SPDEs driven by a multiplicative noise of special structure. We assume that one path of the first N Fourier modes of the solution are continuously observed over a finite time interval, and we derive Bayesian type estimators for the drift coefficient. As custom for Bayesian statistics, we prove a Bernstein-Von Mises theorem for the posterior density, and consequently, we derive some asymptotic properties of the proposed estimators, as N goes to infinity.

In the second part of the talk we will study parameter estimation problems for discretely sampled SPDEs.  We will discuss 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, that consequently are conveniently applied to SPDEs. Both the drift and the volatility coefficients are estimated using two sampling schemes - observing the solution at a fixed time and on a discrete spatial grid, and at a fixed space point and at discrete time instances of a finite interval.

The theoretical results will be illustrated via numerical examples.