Optimal Investment in Incomplete Markets When Wealth May Become Negative

This paper accompanies a previous one by D. Kramkov and the present author. While in [17] we considered utility functions U: R+→R satisfying the Inada conditions U'(0) = ∞ and U'(∞) = 0, in the present paper we consider utility functions$U: \mathbb{R}\rightarrow\mathbb{R}$which are finitely valued, for all x ∈R, and satisfy U'(-∞) = ∞ and U'(∞) = 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 +∞, was isolated: lim sup$_{x\rightarrow +\infty}\frac{xU'(x)}{U(x)} < 1$. 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 -∞, namely, lim inf$_{x\rightarrow -\infty}\frac{xU'(x)}{U(x)} > 1$. 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 : R→R, which is finitely valued on all of 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.