This paper studies an equilibrium model between an insurance buyer and an insurance seller, where both parties' risk preferences are given by convex risk measures. The interaction is modeled through a Stackelberg type game, where the insurance seller plays first by offering prices, in the form of safety loadings. Then the insurance buyer chooses his optimal proportional insurance share and his optimal prevention effort in order to minimize his risk measure. The loss distribution is given by a family of stochastically ordered probability measures, indexed by the prevention effort. We give special attention to the problems of self-insurance and self-protection. We prove that the formulated game admits a unique equilibrium, that we can explicitly solve by further specifying the agents criteria and the loss distribution. In self-insurance, we consider also an adverse selection setting, where the type of the insurance buyers is given by his loss probability, and study the screening and shutdown contracts. Finally, we provide case studies in which we explicitly apply our theoretical results.

This paper deals with an optimal linear insurance demand model, where the protection buyer can also exert time-dynamic costly prevention effort to reduce her risk exposure. This is expressed as a stochastic control problem, that consists in maximizing an exponential utility of a terminal wealth. We assume that the effort reduces the intensity of the jump arrival process, and we interpret this as dynamic self-protection. We solve the problem using a dynamic programming principle approach, and we provide a representation of the certainty equivalent of the buyer as the solution to an SDE. Using this representation, we prove that an exponential utility maximizer has an incentive to modify her effort dynamically only in the presence of a terminal reimbursement in the contract. Otherwise, the dynamic effort is actually constant, for a class of Compound Poisson loss processes. If there is no terminal reimbursement, we solve the problem explicitly and we identify the dynamic certainty equivalent of the protection buyer. This shows in particular that the Lévy property is preserved under exponential utility maximization. We also characterize the constant effort as a the unique minimizer of an explicit Hamiltonian, from which we can determine the optimal effort in particular cases. Finally, after studying the dependence of the SDE associated to the insurance buyer on the linear insurance contract parameter, we prove the existence of an optimal linear cover, that is not necessarily zero or full insurance.