Infinite-dimensional optimization and Bayesian nonparametric learning of stochastic differential equations
This work addresses the challenge of modeling complex stochastic processes for applications in fields like finance and physics, offering a novel integration of optimization and Bayesian methods.
The paper tackles the problem of learning drift functions in stochastic differential equations by developing a systematic Bayesian nonparametric approach that integrates infinite-dimensional optimization results with a hierarchical framework, achieving accurate learning with uncertainty quantification and low-cost sparse learning.
The paper has two major themes. The first part of the paper establishes certain general results for infinite-dimensional optimization problems on Hilbert spaces. These results cover the classical representer theorem and many of its variants as special cases and offer a wider scope of applications. The second part of the paper then develops a systematic approach for learning the drift function of a stochastic differential equation by integrating the results of the first part with Bayesian hierarchical framework. Importantly, our Baysian approach incorporates low-cost sparse learning through proper use of shrinkage priors while allowing proper quantification of uncertainty through posterior distributions. Several examples at the end illustrate the accuracy of our learning scheme.