A Continuous-Time Model of Self-Protection

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.