The asymptotic elasticity of utility functions and optimal investment in incomplete markets.(English)Zbl 0967.91017

The paper is devoted to the problem of maximizing the expected utility of terminal wealth in the frame of a general incomplete semimartingale of a financial market. The used market model consists of a finite number of stocks and one bond. The dynamics of the assets is a stochastic semimartingale process and the price of the bond is constant. There is an economic agent in the model which has a utility function $$U(x)$$ depending on the value of assets in a self-financing portfolio $$x \geq 0$$. The value process $$X = (X_{t} :0 \leq t \leq T)$$ is given by a linear stochastic equation defined by the stocks price process. The goal of the agent is to maximize the expected value from terminal wealth $$EU(X_{T})$$ provided that the initial value of the portfolio $$X_{0} = x$$. The value function of the problem (supremum of $$EU\left( {X_{T}} \right)$$ on all admissible processes) is $$u(x)$$.
The authors’ main questions are: Does the value function $$u(x)$$ exist and is it again a utility function satisfying, as well as $$U(x)$$, the usual assumptions? How can one calculate $$u(x)$$ and the optimal process $$X$$? The main result of the paper is: A necessary and sufficient condition for a positive answer to this questions is the requirement that the asymptotic elasticity of the utility function $$U(x)$$ (the definition is introduced) is strictly less than 1. The investigation and the proposed method for calculating the value function $$u(x)$$ are based on the Legendre transformation.

MSC:

 91G10 Portfolio theory 60G48 Generalizations of martingales 60G44 Martingales with continuous parameter 91B16 Utility theory
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