Topics in Dynamic Programming

Abstract

Dynamic programming is one of the central tools of solving sequential decision problems in economics, finance, management, and other related disciplines. At the same time, dynamic programming underlies computational methods in operations research, artificial intelligence, and reinforcement learning, which increasingly overlap with economics in both methodology and application. Despite its many successes, the core theory of dynamic programming faces two challenges in economic models: the increasing complexity of recursive preferences and discounting structures, and the high dimensionality of realistic economic problems.

From an order-theoretical perspective, this thesis addresses these challenges by embedding the theoretical results in an abstract dynamic programming framework (as introduced in Chapter 2). In particular, Chapter 3 develops new theoretical tools for a single dynamic program in the setting of ordered vector spaces. Chapter 4 provides an important application of Chapter 3 by turning to optimal stopping models and restricting to Banach lattice setting. Chapter 5 characterizes a transformation framework between a high-dimensional dynamic program and a low-dimensional dynamic program, under which their optimality results can be transformed exactly.