Optimal insurance under multiple sources of risk with positive dependence

In this paper we try to derive an optimal insurance treaty when the insured faces multiple sources of risk. We show that the deductible insurance is optimal when the insurable and uninsurable risks are positively dependent or independent within the expected utility framework. A necessary condition of optimal deductible is given under some mild conditions. We compare our model with the classical one without background risk. Furthermore, the shifts of optimal deductible and expected utility by modifications of the dependence structure and the marginal are analyzed.