We provide the solution to a fusion of two fundamental problems in mathematical finance. The first problem is that of maximizing the expected utility of terminal wealth of an investor who holds a short position in a contingent claim, and the second is that of maximizing terminal wealth where the utility function allows the investor to have negative wealth. Under assumptions of reasonable asymptotic elasticity on the investor's utility function, we present an optimal investment theorem and simultaneously treat the corresponding dual problem.
"Utility based optimal hedging in incomplete markets." Ann. Appl. Probab. 12 (2) 691 - 709, May 2002. https://doi.org/10.1214/aoap/1026915621