A continuous-time model of self-protection

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