Cone-Constrained Continuous-Time Markowitz Problems

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 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 ± 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.

Convex Duality in Mean-Variance Hedging Under Convex Trading Constraints

“…Therefore, the existence of a solution to (4.1) follows from Theorem 4.1. The dynamic structure of this solution in a general semimartingale framework is established in [7], which generalises the results obtained for a complete Itô process model in Theorem 6.3 of [13].…”

Mean-Variance Policy for Discrete-Time Cone Constrained Markets: Time Consistency in Efficiency and Minimum-Variance Signed Supermartingale Measure

“…Remark 2.3. Theorem 2.1 reveals that the optimal investment policy is a two-piece linear function with respect to the investor's current wealth level and this finding represents the discrete-time counterpart of the result in Czichowsky and Schweizer (2013) for continuous-time. In Section 5, we will also demonstrate that the result in Theorem 2.1 is also an extension of the result in Cui et al (2014) for multiperiod mean-variance formulation with no-shorting constraint.…”

“…In this section, we will use duality theory and dynamic programming to derive the discrete-time efficient mean-variance policy analytically in convex cone constrained markets. We will demonstrate that the optimal mean-variance policy is a two-piece linear function of the current wealth level, which represents an extension of the result in Cui et al (2014) for discrete-time markets under the no-shorting constraint (a special convex cone) and a discretetime counterpart of the policy in Czichowsky and Schweizer (2013).…”

Mean‐variance Policy for Discrete‐time Cone‐constrained Markets: Time Consistency in Efficiency and the Minimum‐variance Signed Supermartingale Measure

“…In this section, we use duality theory and dynamic programming to analytically derive the discrete-time efficient mean-variance policy in convex coneconstrained markets. We demonstrate that the optimal mean-variance policy is a twopiece linear function of the current wealth level, which represents an extension of the result in Cui et al (2014) for discrete-time markets under the no-shorting constraint (a special convex cone) and a discrete-time counterpart of the policy in Czichowsky and Schweizer (2013).…”

“…Constrained dynamic mean-variance portfolio selection problems with various constraints have been attracting increasing attention in the last decade, e.g., Li, Zhou, and Lim (2002), Zhu, Li, and Wang (2004), Bielecki et al (2005), Sun and Wang (2006), Labbé and Heunis (2007), and Czichowsky and Schweizer (2010). Recently, Czichowsky and Schweizer (2013) further considered cone-constrained continuous-time mean-variance portfolio selection with semimartingale price processes. REMARK 2.2.…”