Optimal Investment in Incomplete Financial Markets
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.
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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 nancial market. Emphasis will be given to the duality theory related to this convex optimization problem. For expository reasons we rst consider the classical case where the underlying probability space Ω is nite. This setting has the advantage that the technical diculties 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 nding 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 nancial 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 nding the optimal investment, as well as for a proper duality theory. It turns out that one may give a necessary and sucient 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 nancial 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. We state two general existence and duality results pertaining to the setting of optimizing expected utility of terminal consumption. We also survey some of the rami cations of these results allowing for intermediate consumption, state-dependent utility, random endowment, non-smooth utility functions and transaction costs.
Support by the Austrian Science Foundation FWF under the Wittgenstein-Preis program Z36-MAT and grant SFB#010 and by the Austrian National Bank under grant 'Jubilumsfondprojekt Number 8699' is gratefully acknowledged.
H. Geman et al. (eds.), Mathematical Finance — Bachelier Congress 2000 © Springer-Verlag Berlin Heidelberg 2002
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Walter Schachermayer
Key words
Optimal Portfolios, Incomplete Markets, Replicating Portfolios, No-arbitrage bounds, Utility Maximization, Asymptotic Elasticity of Utility Functions. JEL classication: C 60, C 61, G 11, G 12, G 13 1 Introduction
A basic problem of mathematical nance is the problem of an economic agent, who invests in a nancial market so as to maximize the expected utility of her terminal
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