The set-valued Bellman's principle: Methodology, applications and computation

Dynamic programming and the Bellman equation have a variety of applications not only in finance, but in many other scientific fields as well. While multivariate problems of optimal control have been solved via dynamic programming in the past, an explicit form of the Bellman equation has not been available until now. We use the lattice approach to vector optimization to derive it. As it is based on a set optimization notion of infimum, we call it a set-valued Bellman principle. We study this principle in the context of the mean-risk problem, treated as a vector optimization problem. As such, the mean-risk problem turns out to be time consistent (in a manner appropriate for a vector optimization problem) and to satisfy the set-valued Bellman principle. These results also find an application in performance maximization when the dynamic risk-adjusted return on capital and the dynamic gain-loss ratio are used to measure performance. Specifically, a time consistent bi-objective (mean-risk or mean-loss) problem can be used to solve a time inconsistent scalar (performance maximization) problem.

Practical computation of the set-valued Bellman principle requires us to solve parametrized vector optimization problems. These can often be reformulated as projection problems. In the polyhedral case, a connection between polyhedral projection and vector linear programs is known. In the final project of this thesis we investigate the connection between convex projection and convex vector programming.