## Super-replication and utility maximization in large financial markets.(English)Zbl 1081.60051

The aim of the paper is to investigate large (with countably many assets) financial markets (and, in particular, the factor models) by considering the problems of super-replication and utility maximization. For this purpose, a theory of stochastic integration with respect to a sequence of semimartingales is considered. A process is called a generalized strategy (as opposed to elementary strategy) if it is integrable with respect to the whole semimartingale sequence. The class of admissible strategies is described and the main super-replication result is formulated in terms of equivalent martingale measures. The dual characterization of the super-replication price paves the way to an extension of the convex duality approach in order to study the utility maximization problem in a large market.
It is proved that under suitable assumptions on the utility function, the problem of the utility maximization has an optimal solution and in some cases it is possible to give an explicit characterization of it. Contrary to the case of the super-replication price, it is established that the supremum of the expected utilities over all elementary strategies coincides with the supremum over the generalized strategies. The last section of the paper is devoted to the analysis of the infinite-dimensional factor models. It is shown that in the large market there is an explicit characterization of the solution of utility maximization problem.

### MSC:

 60H30 Applications of stochastic analysis (to PDEs, etc.) 91G10 Portfolio theory 60H05 Stochastic integrals 60G48 Generalizations of martingales

### Keywords:

equivalent martingale measure; generalized strategy
Full Text:

### References:

 [1] Björk, T.; Näslund, B., Diversified portfolios in continuous time, Europ. fin. rev., 1, 361-387, (1998) · Zbl 1030.91028 [2] W. Brannath, W. Schachermayer, A bipolar theorem for $$L_+^0(\Omega, \mathcal{F}, P)$$, in: Séminaire de Probabilités, XXXIII, Lecture Notes in Mathematics, vol. 1709, Springer, Berlin, 1999, pp. 349-354. [3] De Donno, M., A note on completeness in large financial markets, Math. finance, 14, 2, 295-315, (2004) · Zbl 1090.91032 [4] M. De Donno, M. Pratelli, Stochastic integration with respect to a sequence of semimartingales, Séminaire de Probabilités, XXXIX, 2002, to appear. [5] Delbaen, F.; Schachermayer, W., A general version of the fundamental theorem of asset pricing, Math. ann., 300, 3, 463-520, (1994) · Zbl 0865.90014 [6] El Karoui, N.; Quenez, M.C., Dynamic programming and pricing of contingent claims in an incomplete market, SIAM J. control optim., 33, 1, 29-66, (1995) · Zbl 0831.90010 [7] M. Emery, Une topologie sur l’espace des semimartingales, in: Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78), Lecture Notes in Mathematics, vol. 721, Springer, Berlin, 1979, pp. 260-280. [8] Huberman, G., A simple approach to arbitrage pricing theory, J. econom. theory, 28, 183-191, (1982) · Zbl 0519.90017 [9] Yu.M. Kabanov, On the FTAP of Kreps-Delbaen-Schachermayer, in: Statistics and Control of Stochastic Processes (Moscow, 1995/1996), World Science Publishing, River Edge, NJ, 1997, pp. 191-203. · Zbl 0926.91017 [10] Kabanov, Yu.M.; Kramkov, D.O., Large financial markets: asymptotic arbitrage and contiguity, Theory probab. appl., 39, 1, 222-229, (1994) · Zbl 0834.90018 [11] Kabanov, Yu.M.; Kramkov, D.O., Asymptotic arbitrage in large financial markets, Finance stochast., 2, 2, 143-172, (1998) · Zbl 0894.90020 [12] Klein, I., A fundamental theorem of asset pricing for large financial markets, Math. finance, 10, 4, 443-458, (2000) · Zbl 1034.91042 [13] Klein, I.; Schachermayer, W., Asymptotic arbitrage in non-complete large financial markets, Theory probab. appl., 41, 4, 927-934, (1996) · Zbl 0898.60053 [14] Kramkov, D.O.; Schachermayer, W., The asymptotic elasticity of utility functions and optimal investment in incomplete markets, Ann. appl. probab., 9, 3, 904-950, (1999) · Zbl 0967.91017 [15] D.O. Kramkov, W. Schachermayer, Necessary and sufficient conditions in the problem of optimal investment in incomplete markets, Ann. Appl. Probab. 13 (4) (2003) 1504-1516. · Zbl 1091.91036 [16] Mémin, J., Espaces de semi martingales et changement de probabilité, Z. wahrsch. verw. gebiete, 52, 1, 9-39, (1980) · Zbl 0407.60046 [17] R. Mikulevicius, B.L. Rozovskii, Normalized stochastic integrals in topological vector spaces, in: Séminaire de Probabilités, XXXII, Lecture Notes in Mathematics, vol. 1686, Springer, Berlin, 1998, pp. 137-165. · Zbl 0910.60041 [18] Ross, S., The arbitrage theory of asset pricing, J. econom. theory, 13, 341-360, (1976) [19] W. Schachermayer, Portfolio Optimization in Incomplete Financial Markets, Cattedra Galileiana, Scuola Normale Superiore, Pisa, 2004. · Zbl 1104.91042 [20] R. Tyrrell Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, NJ, 1970. · Zbl 0193.18401
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.