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: R R, which are finitely valued, for all x 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 supₗ+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.