## The Annals of Applied Probability

### Necessary and sufficient conditions in the problem of optimal investment in incomplete markets

#### Abstract

Following Ann. Appl. Probab. 9 (1999) 904--950 we continue the study of the problem of expected utility maximization in incomplete markets. Our goal is to find minimal conditions on a model and a utility function for the validity of several key assertions of the theory to hold true. In the previous paper we proved that a minimal condition on the utility function alone, that is, a minimal market independent condition, is that the asymptotic elasticity of the utility function is strictly less than 1. In this paper we show that a necessary and sufficient condition on both, the utility function and the model, is that the value function of the dual problem is finite.

#### Article information

Source
Ann. Appl. Probab., Volume 13, Number 4 (2003), 1504-1516.

Dates
First available in Project Euclid: 25 November 2003

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1069786508

Digital Object Identifier
doi:10.1214/aoap/1069786508

Mathematical Reviews number (MathSciNet)
MR2023886

Zentralblatt MATH identifier
1091.91036

Subjects
Primary: 90A09 90A10
Secondary: 90C26: Nonconvex programming, global optimization

#### Citation

Kramkov, D.; Schachermayer, W. Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 13 (2003), no. 4, 1504--1516. doi:10.1214/aoap/1069786508. https://projecteuclid.org/euclid.aoap/1069786508

#### References

• Bismut, J. M. (1973). Conguate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44 384--404.
• Cox, J. C. and Huang, C. F. (1989). Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. Math. Econom. 49 33--83.
• Cox, J. C. and Huang, C. F. (1991). A variational problem arising in financial economics. J. Math. Econom. 20 465--487.
• Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463--520.
• Delbaen, F. and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 215--250.
• He, H. and Pearson, N. D. (1991). Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite-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.
• 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 and Optim. 29 702--730.
• Kramkov, D. O. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9 904--950.
• Pliska, S. R. (1986). A stochastic calculus model of continuous trading: Optimal portfolio. Math. Oper. Res. 11 371--382.
• Rockafellar, R. T. (1970). Convex Analysis. Princeton Univ. Press.
• Schachermayer, W. (2001). Optimal investment in incomplete markets when wealth may become negative. Ann. Appl. Probab. 11 694--734.