The Annals of Applied Probability

On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets

Dmitry Kramkov and Mihai Sîrbu

Full-text: Open access

Abstract

We study the two-times differentiability of the value functions of the primal and dual optimization problems that appear in the setting of expected utility maximization in incomplete markets. We also study the differentiability of the solutions to these problems with respect to their initial values. We show that the key conditions for the results to hold true are that the relative risk aversion coefficient of the utility function is uniformly bounded away from zero and infinity, and that the prices of traded securities are sigma-bounded under the numéraire given by the optimal wealth process.

Article information

Source
Ann. Appl. Probab., Volume 16, Number 3 (2006), 1352-1384.

Dates
First available in Project Euclid: 2 October 2006

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1159804984

Digital Object Identifier
doi:10.1214/105051606000000259

Mathematical Reviews number (MathSciNet)
MR2260066

Zentralblatt MATH identifier
1149.91035

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

Keywords
Utility maximization incomplete markets Legendre transformation duality theory risk aversion risk tolerance

Citation

Kramkov, Dmitry; Sîrbu, Mihai. On the two-times differentiability of the value functions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 16 (2006), no. 3, 1352--1384. doi:10.1214/105051606000000259. https://projecteuclid.org/euclid.aoap/1159804984


Export citation

References

  • Delbaen, F. and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 215–250.
  • Gourieroux, C., Laurent, J. P. and Pham, H. (1998). Mean-variance hedging and numéraire. Math. Finance 8 179–200.
  • Harrison, J. M. and Pliska, S. R. (1983). A stochastic calculus model of of continuous trading: Complete markets. Stochastic Process. Appl. 15 313–316.
  • Hiriart-Urruty, J. and Lemaréchal, C. (1993). Convex Analysis and Minimization Algorithms. Springer, Berlin.
  • Jacka, S. D. (1992). A martingale representation result and an application to incomplete financial markets. Math. Finance 2 239–250.
  • Kallsen, J. (2004). $\sigma$-localization and $\sigma$-martingales. Theory Probab. Appl. 48 152–163.
  • Kramkov, D. and Schachermayer, W. (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9 904–950.
  • 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.
  • Kramkov, D. and Sîrbu, M. (2006). The sensitivity analysis of utility based prices and the risk-tolerance wealth processes. Ann. Appl. Probab. To appear.
  • Krylov, N. V. (1980). Controlled Diffusion Processes. Springer, New York.