This paper accompanies a previous one by D.Kramkov and the present
author. While in [17 ] we considered utility functions $U : \mathbb{R}_+ \to
\mathbb{R}$ satisfying the Inada conditions $U'(0) = \infty$ and $U'(\infty) =
0$, in the present paper we consider utility functions $U : \mathbb{R} \to
\mathbb{R}$, which are finitely valued, for all $x\epsilon\mathbb{R}$ and
satisfy $U'(-\infty) = \infty$ and $U'(\infty) = 0. A typical example of this
situation is the exponential utility $U(x) = -e^{-x}$.
¶ In the setting of [17 ] the following crucial condition on the
asymptotic elasticity of U, as x tends to $+\infty$, was
isolated: $lim sup_{x\to+\infty}\frac{xU'(x)}{U(x)}<1$. This condition was
found to be necessary and sufficient for the existence of the optimal
investment as well as other key assertions of the related duality theory to
hold true, if we allow for general semi-martingales to model a (not necessarily
complete) financial market.
¶ In the setting of the present paper this condition has to be
accompanied by a similar condition on the asymptotic elasticity of U, as
x tends to $-\infty$, namely, $\lim
\inf_{x\to-\infty\}frac{xU'(x)}{U(x)}>1$. If both conditions are satisfied
—we then say that the utility function U has reasonable
asymptotic elasticity —we prove an existence theorem for the optimal
investment in a general locally bounded semi-martingale model of a financial
market and for a utility function $U : \mathbb{R} \to \mathbb{R}$, which is
finitely valued on all of $\mathbb{R}$; this theorem is parallel to the main
result of [17 ].We also give examples showing that the reasonable asymptotic
elasticity of U also is a necessary condition for several key assertions
of the theory to hold true.