Cone-Constrained Continuous-Time Markowitz Problems

Christoph Czichowsky and Martin Schweizer
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Christoph Czichowsky: Vienna University of Technology
Martin Schweizer: ETH Zürich

No 12-25, Swiss Finance Institute Research Paper Series from Swiss Finance Institute

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 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 (search for similar items in EconPapers)
JEL-codes: C61 G11 (search for similar items in EconPapers)
Pages: 44 pages
Date: 2012-06
New Economics Papers: this item is included in nep-ore
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Persistent link: https://EconPapers.repec.org/RePEc:chf:rpseri:rp1225

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