Collaborative Research: Ergodic Control of Stochastic Differential Equations Driven By a Class of Pure-Jump Levy Processes, and Applications to Stochastic Networks

Queueing processes in large-scale networks are ubiquitous in society as exemplified by patient flow in hospitals, telephone call centers, service networks, and manufacturing and service operations management. This research will provide fundamental theoretical understanding of these queueing processes in the presence of batch arrivals. The results of this research will be put into direct applications to real-world stochastic networks through collaborations with the health systems and system analysis industry. This research will support under-represented minority groups and train young STEM graduates with new mathematical skills.

This research concerns ergodic control problems for systems described by Ito stochastic differential equations (SDEs) driven by pure-jump Levy processes. The research objectives are: (1) to develop a comprehensive theoretical framework for ergodic control for a large class of controlled SDEs driven by a pure-jump Levy process; (2) to study a novel fully nonlinear problem that arises in admission control and falls outside the usual paradigm of stochastic control; and (3) to develop value iteration algorithms and spatial approximation methods in order to study large time asymptotics, and also to facilitate the numerical solution of the associated Hamilton-Jacobi-Bellman (HJB) equations, and the design of learning schemes for adaptive control in the presence of unknown parameters. This research will advance the basic science of applied mathematics and stochastic control, and make fundamental contributions to applied probability and stochastic networks.