摘要：

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. We then pass to the general case of an arbitrage-free financial market modeled by an 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 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 ramifications of these results allowing for intermediate consumption, state-dependent, utility, random endowment,, non-smooth utility functions and transaction costs.

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