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.


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