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at Wien The paper studies the problem of maximizing the expected utility of terminal wealth in the framework of a general incomplete semimartingale model of a financial market. We show that the necessary and sufficient condition on a utility function for the validity of several key assertions of the theory to hold true is the requirement that the asymptotic elasticity of the utility function is strictly less than 1. 1. Introduction. A basic problem of mathematical finance is the problem of an economic agent who invests in a financial market so as to maximize the expected utility of his terminal wealth. In the framework of a continuous-time model the problem was studied for the first time by Merton in two seminal papers, (27) and (28) (see also (29) as well as (32) for a treatment of the discrete-time case). Using the methods of stochastic optimal control, Merton derived a nonlinear partial differential equation (Bellman equation) for the value function of the optimization problem. He also produced the closed-form solution of this equation, when the utility function is a power function, the logarithm or of the form 1 − e−ηx for some positive η. The Bellman equation of stochastic programming is based on the require- ment of Markov state processes. The modern approach to the problem of ex- pected utility maximization, which permits us to avoid the assumption of Markov asset prices, is based on duality characterizations of portfolios pro- vided by the set of martingale measures. For the case of a complete financial market, where the set of martingale measures is a singleton, this "martin- gale" methodology was developed by Pliska (30), Cox and Huang  4 5 and Karatzas, Lehoczky and Shreve (22). It was shown that the marginal utility of the terminal wealth of the optimal portfolio is, up to a constant, equal to the density of the martingale measure; this key result naturally extends the classical Arrow-Debreu theory of an optimal investment derived in a one-step, finite probability space model. Considerably more difficult is the case of incomplete financial models. It was studied in a discrete-time, finite probability space model by He and Pearson (16) and in a continuous-time diffusion model by He and Pearson (17) and by Karatzas, Lehoczky, Shreve and Xu (21). The central idea here is to solve a

The asymptotic elasticity of utility functions and optimal investment in incomplete markets