Topics in Stochastic Control: Finance, Epidemics, and Machine Learning

This project, consisting of four main topics, aims to create integrated knowledge across mathematical finance, mathematical epidemiology, and machine learning. Topic 1 explores a new method to resolving time inconsistency in optimization. For example, long-term financial planning in a society must confront time inconsistency (as different generations may not agree on an optimal financial planning strategy).

The fixed-point approach to be developed will provide a convenient technical tool for policymakers to find equilibria between generations, or strategies acceptable to all generations. Topic 2 integrates economic analysis into traditional epidemic modeling. It will capture how an epidemic alters individuals' behaviors and how this change of behaviors ultimately influences the epidemic's evolution.

The aim is to facilitate policymaking that anticipates people's reactions to an epidemic. Topic 3 approaches student loans from two complementary angles: how the debt accumulates over a student's years of study and how to repay the debt in a cost-efficient way. This study aims to provide individual borrowers with real savings and policymakers with concrete quantitative tools.

Topic 4 devises new types of gradient flows to strengthen techniques in machine learning. It will provide rigorous mathematical foundations for the design of algorithms and more flexibility to accommodate unknown dynamics. Undergraduate and graduate students will be involved in this project.

The project will develop the fixed-point approach (Topic 1) by merging theory for stochastic flows of diffeomorphisms with convergence theory for stochastic processes. Such a link will give new convergence results for functionals of controlled diffusions and would allow equilibrium controls to be characterized as fixed points of an operator and conveniently found via fixed-point iterations.

Behavioral models for epidemics (Topic 2) rely on a three-population model of consumption behaviors of the susceptible, infected, and recovered. The associated Hamilton-Jacobi-Bellman (HJB) equation involves unusual nonlinearity due to controllable jumps, which will be approached by a combination of viscosity solutions techniques. A student's debt accumulation and optimal repayment (Topic 3) will be investigated through a mean field game whose Hamiltonian may not admit a maximizer and a random-horizon control problem with a stochastically evolving constraint.

Resolving them will demand a vanishing viscosity method based on generalized solutions to a mean field game system and a random-horizon stochastic Pontryagin maximum principle. New types of gradient flows (Topic 4) will be driven by (i) a Langevin-type McKean-Vlasov stochastic differential equation (SDE) or (ii) a coupled system of a Langevin SDE and a controlled diffusion.

Interconnections among SDEs, nonlinear Fokker-Planck equations, and HJB equations will be investigated to uncover the gradient flows' invariant distributions. This will facilitate new stochastic gradient descent approaches to both static and dynamic optimization in the space of probability measures.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.