Robust utility maximisation with intractable claims

We study a continuous-time expected utility maximisation problem where the investor at maturity receives the value of a contingent claim in addition to the investment payoff from the financial market. The investor knows nothing about the claim other than its probability distribution; hence the name "intractable claim". In view of the lack of necessary information about the claim, we consider a robust formulation to maximise her utility in the worst scenario. We apply the quantile formulation to solve the problem, express the quantile function of the optimal terminal investment income as the solution of certain variational inequalities of ordinary differential equations, and obtain the resulting optimal trading strategy. In the case of exponential utility, the problem reduces to a (non-robust) rank-dependent utility maximisation with probability distortion whose solution is available in the literature. The results can also be used to determine the utility indifference price of the intractable claim.