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Optimal investment in incomplete financial markets

Schachermayer, Walter (2002) Optimal investment in incomplete financial markets. Report Series SFB "Adaptive Information Systems and Modelling in Economics and Management Science", 66. SFB Adaptive Information Systems and Modelling in Economics and Management Science, WU Vienna University of Economics and Business, Vienna.

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Abstract

We give a review of classical and recent results on maximization of expected utility for an investor who has the possibility of trading in a financial market. Emphasis will be given to the duality theory related to this convex optimization problem. For expository reasons we first consider the classical case where the underlying probability space is finite. This setting has the advantage that the technical diffculties of the proofs are reduced to a minimum, which allows for a clearer insight into the basic ideas, in particular the crucial role played by the Legendre-transform. In this setting we state and prove an existence and uniqueness theorem for the optimal investment strategy, and its relation to the dual problem; the latter consists in finding an equivalent martingale measure optimal with respect to the conjugate of the utility function. We also discuss economic interpretations of these theorems. We then pass to the general case of an arbitrage-free financial market modeled by an R^d-valued semi-martingale. In this case some regularity conditions have to be imposed in order to obtain an existence result for the primal problem of finding the optimal investment, as well as for a proper duality theory. It turns out that one may give a necessary and sufficient condition, namely a mild condition on the asymptotic behavior of the utility function, its so-called reasonable asymptotic elasticity. This property allows for an economic interpretation motivating the term "reasonable". The remarkable fact is that this regularity condition only pertains to the behavior of the utility function, while we do not have to impose any regularity conditions on the stochastic process modeling the financial market (to be precise: of course, we have to require the arbitrage-freeness of this process in a proper sense; also we have to assume in one of the cases considered below that this process is locally bounded; but otherwise it may be an arbitrary R^d-valued semi-martingale). (author's abstract)

Item Type: Paper
Keywords: optimal portfolios / incomplete markets / replicating portfolios / no-arbitrage bounds / utility maximization / asymptotic elasticity of utility functions
Classification Codes: JEL C60, C61, G11, G12, G13
Divisions: Departments > Finance, Accounting and Statistics > Statistics and Mathematics
Departments > Informationsverarbeitung u Prozessmanag. > Informationswirtschaft
Departments > Informationsverarbeitung u Prozessmanag. > Produktionsmanagement > Taudes
Departments > Marketing > Service Marketing und Tourismus
Depositing User: Repository Administrator
Date Deposited: 25 Mar 2002 12:02
Last Modified: 16 Jun 2011 10:40
URI: http://epub.wu.ac.at/id/eprint/648

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