Portfolio optimization with a prescribed terminal wealth distribution

This paper studies a portfolio allocation problem, where the goal is to reach a prescribed wealth distribution at a final time. We study this problem with the tools of optimal mass transport. We provide a dual formulation which is solved with a gradient descent algorithm. This involves solving an associated Hamilton–Jacobi–Bellman and Fokker–Planck equations with a finite difference method. Numerical examples for various prescribed terminal distributions are given, showing that we can successfully reach attainable targets. We then consider adding consumption during the investment process, to take into account distributions that are either not attainable, or sub-optimal.