Cone-constrained continuous-time Markowitz problems. (English) Zbl 1268.91162

The mean-variance portfolio selection problem (Markowitz problem) consists in finding in a financial market a self-financing strategy whose final wealth has maximal mean and minimal variance. In the present paper this problem is studied in continuous time in a general semimartingale model and under some constraints, meaning that each allowed trading strategy is restricted to always lie in a closed cone which might depend on the state and time in a predictable way. As in the unconstrained case, the solution to the Markowitz problem can be obtained by solving the particular mean-variance hedging problem of approximating in \(L^2\) a constant payoff by the terminal gains of a self-financing trading strategy. To get existence of a solution to the latter problem, it is shown first that the space of constrained terminal gains is closed in \(L^2\); this is sufficient if the constraints are in addition convex. The main focus and achievement of the paper is the subsequent structural description of the optimal strategy by its local properties. This is made possible by treating the approximation in \(L^2\) as a problem in stochastic optimal control. By exploiting the quadratic and conic structure of the problem, the authors obtain a decomposition of its value process into a sum with two opportunity processes \(L^\pm\) appearing as coefficients. Using the martingale optimality principle allows them to describe the drift of \(L^\pm\) and from there the optimal strategy locally in feedback form via the pointwise minimizers of two predictable functions. So, 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 \(L^\pm\). Verification results are proved as well. This explains and generalizes all results so far in the literature on the Markowitz problem under cone constraints. The generality of the framework allows us to capture a new behavior of the optimal strategy.


91G10 Portfolio theory
91G80 Financial applications of other theories
93E20 Optimal stochastic control
60G48 Generalizations of martingales
49N10 Linear-quadratic optimal control problems
Full Text: DOI arXiv Euclid


[1] Aliprantis, C. D. and Border, K. C. (2006). Infinite Dimensional Analysis. A Hitchhiker’s Guide , 3rd ed. Springer, Berlin. · Zbl 1156.46001
[2] Bielecki, T. R., Jin, H., Pliska, S. R. and Zhou, X. Y. (2005). Continuous-time mean-variance portfolio selection with bankruptcy prohibition. Math. Finance 15 213-244. · Zbl 1153.91466
[3] Bobrovnytska, O. and Schweizer, M. (2004). Mean-variance hedging and stochastic control: Beyond the Brownian setting. IEEE Trans. Automat. Control 49 396-408. · Zbl 1366.91152
[4] Černý, A. and Kallsen, J. (2007). On the structure of general mean-variance hedging strategies. Ann. Probab. 35 1479-1531. · Zbl 1124.91028
[5] Černý, A. and Kallsen, J. (2008). A counterexample concerning the variance-optimal martingale measure. Math. Finance 18 305-316. · Zbl 1133.91397
[6] Choulli, T., Krawczyk, L. and Stricker, C. (1998). \({\mathcal{E}}\)-martingales and their applications in mathematical finance. Ann. Probab. 26 853-876. · Zbl 0938.60032
[7] Czichowsky, C. and Schweizer, M. (2010). Convex duality in mean-variance hedging under convex trading constraints. NCCR FINRISK Working Paper 667, ETH Zurich. Available at . Adv. in Appl. Probab. · Zbl 1277.60079
[8] Czichowsky, C. and Schweizer, M. (2011). Closedness in the semimartingale topology for spaces of stochastic integrals with constrained integrands. In Séminaire de Probabilités XLIII (C. Donati-Martin et al., eds.). Lecture Notes in Math. 2006 413-436. Springer, Berlin. · Zbl 1225.60073
[9] Delbaen, F. and Schachermayer, W. (1996). The variance-optimal martingale measure for continuous processes. Bernoulli 2 81-105 [Corrections. Bernoulli 2 (1996) 379-380. MR1440275]. · Zbl 0849.60042
[10] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential B. Theory of Martingales. North-Holland Mathematics Studies 72 . North-Holland, Amsterdam. · Zbl 0494.60002
[11] Donnelly, C. (2008). Convex duality in constrained mean-variance portfolio optimization under a regime-switching model. Ph.D. thesis, Univ. Waterloo.
[12] El Karoui, N. (1981). Les aspects probabilistes du contrôle stochastique. In Ninth Saint Flour Probability Summer School- 1979 ( Saint Flour , 1979) (P. L. Hennequin, ed.). Lecture Notes in Math. 876 73-238. Springer, Berlin. · Zbl 0472.60002
[13] Hu, Y. and Zhou, X. Y. (2005). Constrained stochastic LQ control with random coefficients, and application to portfolio selection. SIAM J. Control Optim. 44 444-466 (electronic). · Zbl 1210.93082
[14] Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Math. 714 . Springer, Berlin. · Zbl 0414.60053
[15] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes , 2nd ed. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 288 . Springer, Berlin. · Zbl 1018.60002
[16] Jin, H. and Zhou, X. Y. (2007). Continuous-time Markowitz’s problems in an incomplete market, with no-shorting portfolios. In Stochastic Analysis and Applications (F. E. Benth et al., eds.). Abel Symp. 2 435-459. Springer, Berlin. · Zbl 1151.91516
[17] Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 558-602. · Zbl 1044.60045
[18] Kohlmann, M. and Tang, S. (2002). Global adapted solution of one-dimensional backward stochastic Riccati equations, with application to the mean-variance hedging. Stochastic Process. Appl. 97 255-288. · Zbl 1064.93050
[19] Korn, R. and Trautmann, S. (1995). Continuous-time portfolio optimization under terminal wealth constraints. ZOR-Math. Methods Oper. Res. 42 69-92. · Zbl 0836.90011
[20] Labbé, C. and Heunis, A. J. (2007). Convex duality in constrained mean-variance portfolio optimization. Adv. in Appl. Probab. 39 77-104. · Zbl 1110.93051
[21] Laurent, J. P. and Pham, H. (1999). Dynamic programming and mean-variance hedging. Finance Stoch. 3 83-110. · Zbl 0924.90021
[22] Li, X., Zhou, X. Y. and Lim, A. E. B. (2002). Dynamic mean-variance portfolio selection with no-shorting constraints. SIAM J. Control Optim. 40 1540-1555 (electronic). · Zbl 1027.91040
[23] Mania, M. and Tevzadze, R. (2003). Backward stochastic PDE and imperfect hedging. Int. J. Theor. Appl. Finance 6 663-692. · Zbl 1094.91029
[24] Markowitz, H. (1952). Portfolio selection. J. Finance 7 77-91.
[25] Markowitz, H. M. (2002). Portfolio Selection : Efficient Diversification of Investments , 2nd ed. Blackwell, Oxford.
[26] Mémin, J. (1980). Espaces de semi martingales et changement de probabilité. Z. Wahrsch. Verw. Gebiete 52 9-39. · Zbl 0407.60046
[27] Nutz, M. (2012). The Bellman equation for power utility maximization with semimartingales. Ann. Appl. Probab. 22 363-406. Available at . · Zbl 1239.91165
[28] Protter, P. E. (2005). Stochastic Integration and Differential Equations , 2nd ed. Stochastic Modelling and Applied Probability 21 . Springer, Berlin.
[29] Rockafellar, R. T. (1976). Integral functionals, normal integrands and measurable selections. In Nonlinear Operators and the Calculus of Variations ( Summer School , Univ. Libre Bruxelles , Brussels , 1975) (J. P. Gossez et al., eds.). Lecture Notes in Math. 543 . 157-207. Springer, Berlin. · Zbl 0374.49001
[30] Schweizer, M. (2001). A guided tour through quadratic hedging approaches. In Option Pricing , Interest Rates and Risk Management (E. Jouini et al., eds.) 538-574. Cambridge Univ. Press, Cambridge. · Zbl 0992.91036
[31] Sun, W. G. and Wang, C. F. (2006). The mean-variance investment problem in a constrained financial market. J. Math. Econom. 42 885-895. · Zbl 1153.91565
[32] Xia, J. (2005). Mean-variance portfolio choice: Quadratic partial hedging. Math. Finance 15 533-538. · Zbl 1102.91061
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.