Relaxed Utility Maximization in Complete Markets

For a relaxed investor—one whose relative risk aversion vanishes as wealth becomes large—the utility maximization problem may not have a solution in the classical sense of an optimal payoff represented by a random variable. This nonexistence puzzle was discovered by Kramkov and Schachermayer (1999), who introduced the reasonable asymptotic elasticity condition to exclude such situations. Utility maximization becomes well posed again representing payoffs as measures on the sample space, including those allocations singular with respect to the physical probability. The expected utility of such allocations is understood as the maximal utility of its approximations with classical payoffs—the relaxed expected utility. This paper decomposes relaxed expected utility into its classical and singular parts, represents the singular part in integral form, and proves the existence of optimal solutions for the utility maximization problem, without conditions on the asymptotic elasticity. Key to this result is the Polish space structure assumed on the sample space.