Utility maximization under solvency constraints and unhedgeable risks

We consider the utility maximization problem for an investor who faces a solvency or risk constraint in addition to a budget constraint. The investor wishes to maximize her expected utility from terminal wealth subject to a bound on her expected solvency at maturity. We measure solvency using a solvency function applied to the terminal wealth. The motivation for our analysis is an optimal investment problem where the investor faces a random and non-hedgable liability at maturity. Keywords: Utility maximisation, Risk constraint, Value-at-Risk, Merton problem, Expected shortfall, Tail-Value-at-Ris