Degree
Doctor of Philosophy (PhD)
Department
Department of Mathematics
Document Type
Dissertation
Abstract
Stochastic differential equations (SDEs) are essential for modeling systems influenced by both deterministic dynamics and random fluctuations, with applications in a wide variety of disciplines. This thesis develops a Bayesian framework for nonparametric learning in SDEs, addressing key challenges in inference, particularly when dealing with complex systems and incomplete data. The thesis begins by establishing a theoretical foundation in optimization over Hilbert spaces, including a generalized representer theorem to address infinite-dimensional optimization problems encountered in nonparametric inference. Building on this, we introduce a Bayesian framework with shrinkage priors to learn drift functions from high-frequency data. Bayesian approach incorporates low-cost sparse learning through proper use of shrinkage priors while allowing proper quantification of uncertainty through posterior distributions. We then extend the framework to scenarios involving sparse and noisy data. A novel bridge method is proposed, integrating Sequential Monte Carlo (SMC) techniques to simulate unobserved states and combining it with an Expectation-Maximization (EM) algorithm for efficient nonparameter estimation. The efficiency of the algorithms are validated through numerical experiments.
Date
11-1-2024
Recommended Citation
Zhou, Jinpu, "Learning Problems Related to Stochastic Differential Equations" (2024). LSU Doctoral Dissertations. 6632.
https://repository.lsu.edu/gradschool_dissertations/6632
Committee Chair
Ganguly, Arnab
Included in
Dynamic Systems Commons, Probability Commons, Statistical Models Commons, Statistical Theory Commons