Abstract
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 setvalued Bellman principle. We study this principle in the context of the meanrisk problem, treated as a vector optimization problem. As such, the meanrisk problem turns out to be time consistent (in a manner appropriate for a vector optimization problem) and to satisfy the setvalued Bellman principle. These results also find an application in performance maximization when the dynamic riskadjusted return on capital and the dynamic gainloss ratio are used to measure performance. Specifically, a time consistent biobjective (meanrisk or meanloss) problem can be used to solve a time inconsistent scalar (performance maximization) problem.
Practical computation of the setvalued 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.
Practical computation of the setvalued 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.
Original language  English 

Awarding Institution 

Publication status  Published  2021 