## The Annals of Probability

### Optimal consumption from investment and random endowment in incomplete semimartingale markets

#### Abstract

We consider the problem of maximizing expected utility from consumption in a constrained incomplete semimartingale market with a random endowment process, and establish a general existence and uniqueness result using techniques from convex duality. The notion of "asymptotic elasticity'' of Kramkov and Schachermayer is extended to the time-dependent case. By imposing no smoothness requirements on the utility function in the temporal argument, we can treat both pure consumption and combined consumption--terminal wealth problems in a common framework. To make the duality approach possible, we provide a detailed characterization of the enlarged dual domain which is reminiscent of the enlargement of $\lone$ to its topological bidual $\linfd$, a space of finitely additive measures. As an application, we treat a constrained Itô process market model, as well as a "totally incomplete'' model.

#### Article information

Source
Ann. Probab., Volume 31, Number 4 (2003), 1821-1858.

Dates
First available in Project Euclid: 12 November 2003

https://projecteuclid.org/euclid.aop/1068646367

Digital Object Identifier
doi:10.1214/aop/1068646367

Mathematical Reviews number (MathSciNet)
MR2016601

Zentralblatt MATH identifier
1076.91017

#### Citation

Karatzas, Ioannis; Žitković, Gordan. Optimal consumption from investment and random endowment in incomplete semimartingale markets. Ann. Probab. 31 (2003), no. 4, 1821--1858. doi:10.1214/aop/1068646367. https://projecteuclid.org/euclid.aop/1068646367

#### References

• Bhaskara Rao, K. P. S. and Bhaskara Rao, M. (1983). Theory of Charges. Academic Press, London.
• Brannath, W. and Schachermayer, W. (1999). A bipolar theorem for subsets of $L^0_+(\Omega,\mathcal F,\mathcal P)$. Séminaire de Probabilités XXXIII. Lecture Notes in Math. 1709 349--354. Springer, Berlin.
• Chung, K.-L. (1974). A Course in Probability Theory, 2nd ed. Academic Press, New York.
• 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.
• Cox, J. C. and Huang, C. F. (1991). A variational problem arising in financial economics. J. Math. Econom. 20 465--487.
• Cuoco, D. (1997). Optimal consumption and equilibrium prices with portfolio constraints and stochastic income. J. Econom. Theory 72 33--73.
• Cvitanić, J. and Karatzas, I. (1992). Convex duality in constrained portfolio optimization. Ann. Appl. Probab. 2 767--818.
• Cvitanić, J., Schachermayer, W. and Wang, H. (2001). Utility maximization in incomplete markets with random endowment. Finance Stoch. 5 237--259.
• Delbaen, F. and Schachermayer, W. (1993). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463--520.
• Delbaen, F. and Schachermayer, W. (1995). The existence of absolutely continuous local martingale measures. Ann. Appl. Probab. 5 926--945.
• Delbaen, F. and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 215--250.
• Delbaen, F. and Schachermayer, W. (1999). A compactness principle for bounded sequences of martingales with applications. In Proceedings of the Seminar on Stochastic Analysis, Random Fields and Applications 137--173. Birkhäuser, Basel.
• Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential, Vol. B: Theory of Martingales. North-Holland, Amsterdam.
• El Karoui, N. and Jeanblanc-Picqué, M. (1998). Optimization of consumption with labor income. Finance Stoch. 2 409--440.
• El Karoui, N. and Quenez, M.-C. (1995). Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33 29--66.
• Föllmer, H. and Kramkov, D. (1997). Optional decomposition under constraints. Probab. Theory Related Fields 109 1--25.
• He, H. and Pearson, N. D. (1991). Consumption and portfolio policies with incomplete markets and short-sale constraints: The finite-dimensional case. Math. Finance 1 1--10.
• He, H. and Pearson, N. D. (1991). Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite-dimensional case. J. Econom. Theory 54 259--304.
• Jacod, J. (1979). Calcul stochastique et problèmes de martingales. Lecture Notes in Math. 714. Springer, Berlin.
• Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
• Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Springer, New York.
• 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.
• 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.
• Komlós, J. (1967). A generalization of a problem of Steinhaus. Acta Math. Hungar. 18 217--229.
• Kramkov, D. (1996). Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Related Fields 105 459--479.
• Kramkov, D. and Schachermayer, W. (1999). A condition on the asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9 904--950.
• Lakner, P. and Slud, E. (1991). Optimal consumption by a bond investor: The case of random interest rate adapted to a point process. SIAM J. Control Optim. 29 638--655.
• Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. Review of Economic Statistics 51 247--257.
• Merton, R. C. (1971). Optimum consumption and portfolio rules in a conitinuous-time model. J. Econom. Theory 3 373--413.
• Pliska, S. R. (1986). A stochastic calculus model of continuous trading: Optimal portfolio. Math. Oper. Res. 11 371--382.
• Protter, Ph. (1990). Stochastic Integration and Differential Equations. Springer, New York.
• Rockafellar, R. T. (1970). Convex Analysis. Princeton Univ. Press.
• Schachermayer, W. (2000). Optimal investment in incomplete financial markets. In Proceedings of the First World Congress of the Bachelier Society 427--462. Springer, Berlin.
• Schwartz, M. (1986). New proofs of a theorem of Komlós. Acta Math. Hungar. 47 181--185.
• Shiryaev, A. N. (1996). Probability, 2nd ed. Springer, New York. [Translated from the first (1980) Russian edition by R. P. Boas.]
• Strasser, H. (1985). Mathematical Theory of Statistics. de Gruyter, Berlin.
• Wojatszcyk, P. (1996). Banach Spaces for Analysts. Cambridge Univ. Press.
• Xu, G.-L. (1990). A duality method for optimal consumption and investment under short-selling prohibition. Ph.D. dissertation, Dept. Mathematics, Carnegie-Mellon Univ.
• Yosida, K. and Hewitt, E. (1952). Finitely additive measures. Trans. Amer. Math. Soc. 72 46--66.
• Žitković, G. (2002). A filtered version of the bipolar theorem of Brannath and Schachermayer. J. Theoret. Probab. 15 41--61.