B0431
Title: Nonparametric calibration for stochastic reaction-diffusion equations based on discrete observations
Authors: Florian Hildebrandt - University of Hamburg (Germany) [presenting]
Mathias Trabs - Karlsruhe Institute of Technology (Germany)
Abstract: In view of a growing number of stochastic partial differential equation (SPDE) models used in the natural sciences and mathematical finance, their data-based calibration has become an increasingly active field of research during the last few years. Nonparametric estimation for semilinear SPDEs, namely stochastic reaction-diffusion equations in one space dimension, is discussed. We consider observations of the solution field on a discrete grid in time and space with infill asymptotics in both coordinates. Firstly, based on a precise analysis of the Hoelder regularity of the solution process and its nonlinear component, we deduce that the asymptotic properties of diffusivity and volatility estimators derived from realized quadratic space-time variations in the linear setup generalize to the semilinear SPDE. Doing so, we obtain a rate-optimal joint estimator of the two parameters. Secondly, we present a nonparametric estimator for the reaction function specifying the underlying equation. The estimate is chosen from a finite-dimensional function space based on a least-squares criterion. An oracle inequality with respect to the $L^2$-risk provides conditions for the estimator to achieve the usual nonparametric convergence rate. Adaptivity is provided via model selection.
