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April 2013 Cone-constrained continuous-time Markowitz problems
Christoph Czichowsky, Martin Schweizer
Ann. Appl. Probab. 23(2): 764-810 (April 2013). DOI: 10.1214/12-AAP855


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 characterize and construct optimal strategies. Our results explain and generalize all the results available in the literature so far. Moreover, we even obtain new sharp results in the unconstrained case.


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Christoph Czichowsky. Martin Schweizer. "Cone-constrained continuous-time Markowitz problems." Ann. Appl. Probab. 23 (2) 764 - 810, April 2013.


Published: April 2013
First available in Project Euclid: 12 February 2013

zbMATH: 1268.91162
MathSciNet: MR3059275
Digital Object Identifier: 10.1214/12-AAP855

Primary: 49N10 , 60G48 , 91G10 , 91G80 , 93E20

Keywords: $\mathcal{E}$-martingales , BSDEs , cone constraints , linear-quadratic control , Markowitz problem , martingale optimality principle , Mean-variance hedging , opportunity process , portfolio selection , Semimartingales , Stochastic control

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.23 • No. 2 • April 2013
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