The Annals of Applied Probability

On utility maximization under convex portfolio constraints

Kasper Larsen and Gordan Žitković

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Abstract

We consider a utility-maximization problem in a general semimartingale financial model, subject to constraints on the number of shares held in each risky asset. These constraints are modeled by predictable convex-set-valued processes whose values do not necessarily contain the origin; that is, it may be inadmissible for an investor to hold no risky investment at all. Such a setup subsumes the classical constrained utility-maximization problem, as well as the problem where illiquid assets or a random endowment are present.

Our main result establishes the existence of optimal trading strategies in such models under no smoothness requirements on the utility function. The result also shows that, up to attainment, the dual optimization problem can be posed over a set of countably-additive probability measures, thus eschewing the need for the usual finitely-additive enlargement.

Article information

Source
Ann. Appl. Probab., Volume 23, Number 2 (2013), 665-692.

Dates
First available in Project Euclid: 12 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1360682026

Digital Object Identifier
doi:10.1214/12-AAP850

Mathematical Reviews number (MathSciNet)
MR3059272

Zentralblatt MATH identifier
1262.91129

Subjects
Primary: 91G10: Portfolio theory 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

Keywords
Utility maximization convex constraints semimartingales finitely-additive measures convex duality

Citation

Larsen, Kasper; Žitković, Gordan. On utility maximization under convex portfolio constraints. Ann. Appl. Probab. 23 (2013), no. 2, 665--692. doi:10.1214/12-AAP850. https://projecteuclid.org/euclid.aoap/1360682026


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References

  • [1] Aliprantis, C. D. and Border, K. C. (2006). Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd ed. Springer, Berlin.
  • [2] Ben-Israel, A. and Greville, T. N. E. (2003). Generalized Inverses: Theory and Applications, 2nd ed. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC 15. Springer, New York.
  • [3] Bhaskara Rao, K. P. S. and Bhaskara Rao, M. (1983). Theory of Charges: A Study of Finitely Additive Measures. Pure and Applied Mathematics 109. Academic Press, New York.
  • [4] Bouchard, B., Touzi, N. and Zeghal, A. (2004). Dual formulation of the utility maximization problem: The case of nonsmooth utility. Ann. Appl. Probab. 14 678–717.
  • [5] Brannath, W. and Schachermayer, W. (1999). A bipolar theorem for $L^{0}_{+}(\Omega,{\mathcal{F}},\mathbf{P})$. In Séminaire de Probabilités, XXXIII. Lecture Notes in Math. 1709 349–354. Springer, Berlin.
  • [6] Cherny, A. S. and Shiryaev, A. N. (2002). Vector stochastic integrals and the fundamental theorems of asset pricing. In Proceedings of the Steklov Mathematical Institute 237 12–56. Nauka, Moscow.
  • [7] Cox, J. C. and Huang, C.-f. (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. Econom. Theory 49 33–83.
  • [8] Cuoco, D. (1997). Optimal consumption and equilibrium prices with portfolio constraints and stochastic income. J. Econom. Theory 72 33–73.
  • [9] Cvitanić, J. and Karatzas, I. (1992). Convex duality in constrained portfolio optimization. Ann. Appl. Probab. 2 767–818.
  • [10] Cvitanić, J., Schachermayer, W. and Wang, H. (2001). Utility maximization in incomplete markets with random endowment. Finance Stoch. 5 259–272.
  • [11] Czichowsky, C. and Schweizer, M. (2011). Closedness in the semimartingale topology for spaces of stochastic integrals with constrained integrands. In Séminaire de Probabilités XLIII. Lecture Notes in Math. 2006 413–436. Springer, Berlin.
  • [12] Czichowsky, C., Westray, N. and Zheng, H. (2011). Convergence in the semimartingale topology and constrained portfolios. In Séminaire de Probabilités XLIII. Lecture Notes in Math. 2006 395–412. Springer, Berlin.
  • [13] Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463–520.
  • [14] Delbaen, F. and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 215–250.
  • [15] Föllmer, H. and Kramkov, D. (1997). Optional decompositions under constraints. Probab. Theory Related Fields 109 1–25.
  • [16] Föllmer, H. and Leukert, P. (2000). Efficient hedging: Cost versus shortfall risk. Finance Stoch. 4 117–146.
  • [17] Hugonnier, J. and Kramkov, D. (2004). Optimal investment with random endowments in incomplete markets. Ann. Appl. Probab. 14 845–864.
  • [18] Jacod, J. (1980). Intégrales stochastiques par rapport à une semimartingale vectorielle et changements de filtration. In Seminar on Probability, XIV (Paris, 1978/1979) (French). Lecture Notes in Math. 784 161–172. Springer, Berlin.
  • [19] 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.
  • [20] Kallsen, J. (1999). A utility maximization approach to hedging in incomplete markets. Math. Methods Oper. Res. 50 321–338.
  • [21] Kallsen, J. (2002). Derivative pricing based on local utility maximization. Finance Stoch. 6 115–140.
  • [22] Karatzas, I. and Kardaras, C. (2007). The numéraire portfolio in semimartingale financial models. Finance Stoch. 11 447–493.
  • [23] Karatzas, I., Lehoczky, J. P. and Shreve, S. E. (1987). Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. SIAM J. Control Optim. 25 1557–1586.
  • [24] Karatzas, I., Lehoczky, J. P., Shreve, S. E. and Xu, G.-L. (1991). Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control Optim. 29 702–730.
  • [25] Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Applications of Mathematics (New York) 39. Springer, New York.
  • [26] Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9 904–950.
  • [27] Kramkov, D. and Schachermayer, W. (2003). Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 13 1504–1516.
  • [28] Larsen, K. (2009). Continuity of utility-maximization with respect to preferences. Math. Finance 19 237–250.
  • [29] Long, N.-T. (2004). Investment optimization under constraints. Math. Methods Oper. Res. 60 175–201.
  • [30] Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. Rev. Econom. Statist. 51 247–257.
  • [31] Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3 373–413.
  • [32] Mnif, M. and Pham, H. (2001). Stochastic optimization under constraints. Stochastic Process. Appl. 93 149–180.
  • [33] Owen, M. P. and Žitković, G. (2009). Optimal investment with an unbounded random endowment and utility-based pricing. Math. Finance 19 129–159.
  • [34] Pham, H. (2002). Minimizing shortfall risk and applications to finance and insurance problems. Ann. Appl. Probab. 12 143–172.
  • [35] Pham, H. (2009). Continuous-Time Stochastic Control and Optimization with Financial Applications. Stochastic Modelling and Applied Probability 61. Springer, Berlin.
  • [36] Rockafellar, R. T. (1970). Convex Analysis. Princeton Mathematical Series 28. Princeton Univ. Press, Princeton, NJ.
  • [37] Shiryaev, A. N. (1999). Essentials of Stochastic Finance: Facts, Models, Theory. Advanced Series on Statistical Science and Applied Probability 3. World Scientific, River Edge, NJ. Translated from the Russian manuscript by N. Kruzhilin.
  • [38] Sion, M. (1958). On general minimax theorems. Pacific J. Math. 8 171–176.
  • [39] Siorpaes, P. (2010). The relation between arbitrage free prices and marginal utility based prices. Ph.D. thesis, Carnegie Mellon Univ., Pittsburgh, PA.
  • [40] Westray, N. and Zheng, H. (2009). Constrained nonsmooth utility maximization without quadratic inf convolution. Stochastic Process. Appl. 119 1561–1579.
  • [41] Westray, N. and Zheng, H. (2011). Minimal sufficient conditions for a primal optimizer in nonsmooth utility maximization. Finance Stoch. 15 501–512.
  • [42] Zălinescu, C. (2002). Convex Analysis in General Vector Spaces. World Scientific, River Edge, NJ.
  • [43] Žitković, G. (2005). Utility maximization with a stochastic clock and an unbounded random endowment. Ann. Appl. Probab. 15 748–777.
  • [44] Žitković, G. (2010). Convex compactness and its applications. Math. Financ. Econ. 3 1–12.