Time consistency of the mean-risk problem

Consider the dynamic mean-risk problem. Typically, the problem is scalarized and well known not to satisfy the Bellman principle. Thus, the classical dynamic programming methods are not applicable.We will show that when we do... [ view full abstract ]

Consider the dynamic mean-risk problem. Typically, the problem is scalarized and well known not to satisfy the Bellman principle. Thus, the classical dynamic programming methods are not applicable.

We will show that when we do not scalarize the problem, but leave it in its original form as a vector optimization problem, the upper images, whose boundary is the efficient frontier, recurse backwards in time under very mild assumptions. Thus, the dynamic mean-risk problem does satisfy a Bellman principle, but a more general one.

This opens the door for a new branch in mathematics: dynamic multivariate programming.