Optimal insurance design under belief-dependent utility and ambiguity

We introduce a smooth decision model under ambiguity by the belief-dependent utility (BDU) proposed in Fan (Acta Math Appl Sin 37(4):682–696, 2021). Using the smooth decision model under BDU, we get the explicit optimal insurance policy for the insurer. Then the optimal insurance policy for the insured under premium constraint (the insurer is assumed to be risk neutral) is studied. The explicit results can explain some notable behaviors in insurance demand which are inconsistent with the classical insurance contracting literature. For example, if the insured is very sensitive to small losses and the insurer is not so sensitive to small losses (or the insurer is risk neutral), the insured will prefer to purchase warranties for small losses rather than buy protections against devastating losses, which is consistent with some insurance demand behaviors observed on the insurance market. If the insured is less sensitive to small losses than the insurer, insurance policy against large losses above a deductible will be popular. At last, this paper provides an example.