Mathematical Finance, Stochastic Analysis, and Machine Learning Seminar by Gabriela Kovacova - Set-Valued Bellman's Principle: Applications and Computations

Speaker: Kovacova, Gabriela. Vienna University of Economics and Business.

Title: Set-Valued Bellman's Principle: Applications and Computations.

Abstract: Dynamic programming, introduced by Richard Bellman in 1954, is an essential tool used in engineering, applied mathematics and natural sciences. It allows us to solve a control problem by splitting it into a sequence of smaller subproblems. At its heart lies the famous Bellman's equation (or its PDE counterpart, the Hamilton-Jacobi-Bellman equation). In this talk we present a counterpart of the Bellman's equation appropriate for multi-objective problems -- the set-valued Bellman's equation -- and discuss the computational issues.

In the first part of the talk, we derive the set-valued Bellman's equation for an important problem of financial mathematics: Selecting a portfolio of risky assets which maximizes the expected terminal values at the same time as it minimizes portfolio risk is known as the mean-risk problem. The usual approach in the literature is to combine the mean and the risk to obtain a problem with a single objective. This, however, comes at the cost of time inconsistency.

We use multi-objective and set optimization to handle the problem. This yields not only novel economic insights, but also an appropriate form of the Bellman's equation, which opens doors for multi-objective dynamic programming. 

In the second part of the talk, we address some computational issues. Implementing the set-valued Bellman's equation involves solving multi-objective (vector) optimization problems. The existing methods in the literature cover bounded convex vector optimization problems. We relax the assumption of boundedness and propose a method appropriate for both bounded and unbounded problems.

 

Mathematical Finance, Stochastic Analysis, and Machine Learning Seminar