The earth mover’s distance through the lens of algebraic combinatorics

Mr. Brian Hospital
University of Wisconsin-Milwaukee
Dissertator

Under consideration are convergence results between optimality criteria for two canonical stochastic control problems: the long-term average problem and the α-discounted problem, where α>0 is a given discount rate. The objects under control in these problems are stochastic processes that arise as (relaxed) solutions to a controlled martingale problem; and such controlled processes—subject to a given cost constraint—comprise the feasible sets for each of the control problems. In this dissertation, we define and characterize the expected occupation measures associated with these stochastic control problems, and proceed to reformulate each problem as a linear program over a space of such measures. Our task is then to identify sufficient conditions under which the long-term average linear program can be “asymptotically approximated” by the family of (suitably normalized) α-discounted linear programs as α ↓ 0. This approach is what can be referred to as the vanishing discount method. To state the desired conditions precisely, our analysis then turns to set-valued mappings called correspondences. In particular, once the appropriate framework is established, we are able to present our main results in a manner similar to that of Berge’s Theorem. The dissertation concludes with some basic examples and applications that will provide further insight into the structural relationship between the long-term average and α-discounted problems, and demonstrate how this version of the vanishing discount method may be applied in practice.

Advisor: Prof. Richard Stockbridge

Committee Members:
Profs. Chao Zhu, Kevin McLeod, David Spade, and Peter Hinow

Online via Zoom: bit.ly/3Q0to9R