Cone-Constrained Continuous-Time Markowitz Problems

TL;DR

This paper addresses continuous-time Markowitz portfolio optimization under cone constraints, establishing existence, characterizing optimal strategies via stochastic control, and generalizing previous results with new insights, including in unconstrained scenarios.

Contribution

It provides a comprehensive framework for solving constrained continuous-time Markowitz problems using stochastic control and BSDEs, extending and unifying existing literature.

Findings

Existence of solutions for convex cone constraints.

Characterization of optimal strategies via coupled BSDEs.

Generalization of previous results and new insights in unconstrained cases.

Abstract

The Markowitz problem consists of finding in a financial market a self-financing trading strategy whose final wealth has maximal mean and minimal variance. We study this in continuous time in a general semimartingale model and under cone constraints: Trading strategies must take values in a (possibly random and time-dependent) closed cone. We first prove existence of a solution for convex constraints by showing that the space of constrained terminal gains, which is a space of stochastic integrals, is closed in L^2. Then we use stochastic control methods to describe the local structure of the optimal strategy, as follows. The value process of a naturally associated constrained linear-quadratic optimal control problem is decomposed into a sum with two opportunity processes L^{\pm} appearing as coefficients. The martingale optimality principle translates into a drift condition for the semimartingale characteristics of L^{\pm} or equivalently into a coupled system of backward stochastic differential equations for L^{\pm}. We show how this can be used to both characterise and construct optimal strategies. Our results explain and generalise all the results available in the literature so far. Moreover, we even obtain new sharp results in the unconstrained case.