×

The asymptotic elasticity of utility functions and optimal investment in incomplete markets. (English) Zbl 0967.91017

The paper is devoted to the problem of maximizing the expected utility of terminal wealth in the frame of a general incomplete semimartingale of a financial market. The used market model consists of a finite number of stocks and one bond. The dynamics of the assets is a stochastic semimartingale process and the price of the bond is constant. There is an economic agent in the model which has a utility function \(U(x)\) depending on the value of assets in a self-financing portfolio \(x \geq 0\). The value process \(X = (X_{t} :0 \leq t \leq T)\) is given by a linear stochastic equation defined by the stocks price process. The goal of the agent is to maximize the expected value from terminal wealth \(EU(X_{T})\) provided that the initial value of the portfolio \(X_{0} = x\). The value function of the problem (supremum of \(EU\left( {X_{T}} \right)\) on all admissible processes) is \(u(x)\).
The authors’ main questions are: Does the value function \(u(x)\) exist and is it again a utility function satisfying, as well as \(U(x)\), the usual assumptions? How can one calculate \(u(x)\) and the optimal process \(X\)? The main result of the paper is: A necessary and sufficient condition for a positive answer to this questions is the requirement that the asymptotic elasticity of the utility function \(U(x)\) (the definition is introduced) is strictly less than 1. The investigation and the proposed method for calculating the value function \(u(x)\) are based on the Legendre transformation.

MSC:

91G10 Portfolio theory
60G48 Generalizations of martingales
60G44 Martingales with continuous parameter
91B16 Utility theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ansel, J. P. and Stricker, C. (1994). Couverture des actifs contingents et prix maximum. Ann. Inst. H. Poincaré 30 303-315. · Zbl 0796.60056
[2] Bismut, J. M. (1973). Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44 384-404. · Zbl 0276.93060
[3] Brannath, W. and W. Schachermay er (1999). A bidual theorem for subsets of L0+ P Séminaire de Probabilités.
[4] Cox, J. C. and Huang, C. F. (1989). Optimal consumption and portfolio plicies when asset prices follow a diffusion process. J. Econom. Theory 49 33-83. · Zbl 0678.90011
[5] Cox, J. C. and Huang, C. F. (1991). A variational problem arising in financial economics. J. Math. Econom. 20 465-487. · Zbl 0734.90009
[6] Davis, M. H. A. (1997). Option pricing in incomplete markets. In Mathematics of Derivative Securities 216-226. Cambridge Univ. Press. · Zbl 0914.90017
[7] Delbaen, F. and Schachermay er, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300 463-520. · Zbl 0865.90014
[8] Delbaen, F. and Schachermay er, W. (1995). The no-arbitrage property under a change of numéraire. Stochastics Stochastics Rep. 53 213-226. · Zbl 0857.90007
[9] Delbaen, F. and Schachermay er, W. (1998). A simple counter-example to several problems in the theory of asset pricing, which arises in many incomplete markets. Math. Finance. · Zbl 0910.60038
[10] Delbaen F. and Schachermay er, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 215-250. · Zbl 0917.60048
[11] Diestel, J. (1975). Geometry of Banach Spaces-Selected Topics. Lecture Notes in Math. 485. Springer, Berlin. · Zbl 0307.46009
[12] Foldes, L. P. (1990). Conditions for optimality in the infinite-horizon portfolio-cum-savings problem with semimartingale investments. Stochastics Stochastics Rep. 29 133-171. · Zbl 0694.90038
[13] F öllmer, H., Yu and Kabanov, M. (1998). Optional decomposition and Lagrange multiplies. Finance and Stochastics 2 69-81. · Zbl 0894.90016
[14] F öllmer, H. and Kramkov, D. O. (1997). Optional decompositions under constraints. Probab. Theory Related Fields 109 1-25. · Zbl 0882.60063
[15] Grothendieck, A. (1954). Espaces vectoriel topologiques. Sociedade de Matematica de S ao Paulo. · Zbl 0058.33401
[16] 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. · Zbl 0736.90017
[17] 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. · Zbl 0736.90017
[18] Jacka, S. D. (1992). A martingale representation result and an application to incomplete financial markets. Math. Finance 2 239-250. · Zbl 0900.90044
[19] Jacod, J. (1979). Calcul stochastique et problémes de martingales. Lecture Notes in Math. 714. Springer, Berlin. · Zbl 0414.60053
[20] Karatzas, I. (1989). Optimization problems in the theory of continuous trading. SIAM J. Control Optim. 27 1221-1259. · Zbl 0701.90008
[21] Karatzas, I., Lehoczky, J. P., Shreve, S. E. and Xu, G. L. (1991). Martingale and duality methods for utility maximisation in an incomplete market. SIAM J. Control Optim. 29 702-730. · Zbl 0733.93085
[22] 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. · Zbl 0644.93066
[23] Karatzas, I. and Shreve, S. (1991). Brownian Motion and Stochastic Calculus. Springer, New York. · Zbl 0734.60060
[24] El Karoui, N. and Quenez, M.-C. (1995). Dy namic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33 29-66. · Zbl 0831.90010
[25] Kramkov, D. O. (1996). Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probability Theory Related Fields 105 459-479. · Zbl 0853.60041
[26] Krasnosel’skii, M. A. and Rutickii, Ya. B. (1961). Convex Functions and Orlicz Spaces. Gordon and Breach, New York.
[27] Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econom. Statist. 51 247-257.
[28] Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3 373-413. · Zbl 1011.91502
[29] Merton, R. C. (1990). Continuous-Time Finance. Blackwell, Cambridge. · Zbl 1019.91502
[30] Pliska, S. R. (1986). A stochastic calculus model of continuous trading: optimal portfolio. Math. Oper. Res. 11 371-382. · Zbl 1011.91503
[31] Rockafellar, R. T. (1970). Convex Analy sis. Princeton Univ. Press. · Zbl 0193.18401
[32] Samuelson, P. (1969). Lifetime portfolio selection by the dy namic stochastic programming, Rev. Econom. Stoch. 239-246.
[33] Strasser, H. (1985). Mathematical Theory of Statistics: Statistical Experiments and Asy mptotic. Decision Theory. de Gruy ter, Berlin. · Zbl 0594.62017
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.