Shortfall risk models when information of loss function is incomplete

Utility-based shortfall risk measure (SR) effectively captures decision maker’s risk attitude on tail losses by an increasing convex loss function. In this paper, we consider a situation where the decision maker’s risk attitude towards tail losses is ambiguous and introduce a robust version of SR which mitigates the risk arising from such ambiguity. Specifically, we use some available partial information or subjective judgement to construct a set of utility-based loss functions and define a so-called preference robust SR (PRSR) through the worst loss function from the (ambiguity) set. To implement PRSR in practice, we propose three approaches for constructing the ambiguity set. We then apply the PRSR to optimal decision making problems and demonstrate how the latter can be reformulated as tractable convex programs when the underlying exogenous uncertainty is discretely distributed. In the case when the probability distribution is continuous, we propose a sample average approximation scheme and show that it converges to the true problem in terms of the optimal value and the optimal solutions as the sample size increases.