Cone-Constrained Continuous-Time Markowitz Problems

The Markowitz problem consists of finding in a financial market a self-financingtrading strategy whose final wealth has maximal mean and minimal variance. Westudy this in continuous time in a general semimartingale model and under coneconstraints: Trading strategies must take values in a (possibly random and timedependent)closed cone. We first prove existence of a solution for convex constraintsby showing that the space of constrained terminal gains, which is a space of stochasticintegrals, is closed in L2. Then we use stochastic control methods to describe thelocal structure of the optimal strategy, as follows. The value process of a naturallyassociated constrained linear-quadratic optimal control problem is decomposed intoa sum with two opportunity processes L± appearing as coefficients. The martingaleoptimality principle translates into a drift condition for the semimartingalecharacteristics of L± or equivalently into a coupled system of backward stochasticdifferential equations for L±. We show how this can be used to both characteriseand construct optimal strategies. Our results explain and generalise all the resultsavailable in the literature so far. Moreover, we even obtain new sharp results in theunconstrained case.[...]