PhD Defense Predrag Pilipovic
Title: Statistical Inference for Stochastic Differential Equations using Splitting Schemes
Abstract:
This thesis develops and analyzes advanced parameter estimation techniques for discretely observed nonlinear first- and second-order stochastic differential equations (SDEs), focusing on splitting schemes and their applications.
Initially, new numerical properties of splitting schemes, specifically the Lie-Trotter and Strang schemes, are established, enabling more accurate and robust parameter estimation under less restrictive assumptions on the drift parameter. Theoretical advancements include proving the L p convergence of the Strang splitting scheme and demonstrating the consistency and asymptotic efficiency of the associated estimator, confirmed in a simulation study of the three-dimensional stochastic Lorenz system.
Expanding this work to second-order SDEs, we introduce and adapt the Strang splitting scheme to address hypoelliptic systems and scenarios involving partial observations caused by the unobserved velocity variable. The proposed estimators are shown to be both theoretically robust and computationally fast, with variations in the asymptotic variance depending on the likelihood approach used. The theory is illustrated by applying the Kramers oscillator model to model paleoclimate data.
The thesis further extends to developing multivariate Pearson diffusion models, which generalize existing univariate Pearson diffusion frameworks by incorporating linear drift and a quadratic function in the diffusion structure. The Strang splitting scheme for nonlinear processes with Pearson-type noise is proposed, and the closed-form solutions for the first two moments are derived. The applicability of these models is demonstrated through their appearance in genetic research and epidemiological modeling, as well as a generalization of the Kramers model with the student-type noise. The simulation studies validate the dominance of the proposed estimator in estimating diffusion parameters with higher accuracy compared to existing methods.
Supervisors:
Professor Susanne Ditlevsen,University of Copenhagen
Professor Roland Langrock,University of Bielefeld
Assessment Committee:
Professor Michael Sørensen (chair), University of Copenhagen
Professor Alexandros Beskos, University College London
Professor Frank Riedel, University of Bielefeld