Cone-Constrained Continuous-Time Markowitz Problems

44 Pages Posted: 15 Jun 2012  

Christoph Czichowsky

Vienna University of Technology

Martin Schweizer

ETH Zürich - Department of Mathematics

Date Written: June 4, 2012


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 timedependent) 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 L2. Then we use stochastic controlmethods 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± appearing as coefficients. The martingale optimality principle translates into a drift condition for the semimartingale characteristics of L± or equivalently into a coupled system of backward stochastic differential equations for L±. 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.

Keywords: Markowitz problem, cone constraints, portfolio selection, mean-variance, hedging, stochastic control, semimartingales, BSDEs, martingale optimality principle, opportunity process, E-martingales, linear-quadratic control

JEL Classification: G11, C61

Suggested Citation

Czichowsky, Christoph and Schweizer, Martin, Cone-Constrained Continuous-Time Markowitz Problems (June 4, 2012). Swiss Finance Institute Research Paper No. 12-25. Available at SSRN: or

Christoph Czichowsky

Vienna University of Technology ( email )

Karlsplatz 13

Martin Schweizer (Contact Author)

ETH Zürich - Department of Mathematics ( email )

ETH Zentrum HG G51.2
CH-8092 Zurich

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