Optimal allocation of policy limits and deductibles in a model with mixture risks and discount factors

By maximizing the expected utility, we study the optimal allocation of policy limits and deductibles from the viewpoint of a policyholder, where the dependence structure of losses is unknown. In Cheung (2007) [K.C. Cheung, Optimal allocation of policy limits and deductibles, Insurance: Mathematics and Economics 41 (2007) 382-391], the author had considered similar problems. He supposed that a policyholder was exposed to n random losses, and the losses were general risks there, i.e., the loss on each policy was just a random variable. In this paper, the model is extended in two directions. On one hand, we assume that n policies of the n losses are effected by random environments. For each policy, the loss under a fixed environment is characterized by a random variable, so the loss on each policy is a mixture of some fundamental random variables. On the other hand, loss frequencies, which are stochastic, are also considered. Therefore, the whole model is equipped with mixture risks and discount factors. Finally, we get the orderings of the optimal allocations of policy limits and deductibles. Our conclusions also extend the main results in Hua and Cheung (2008) [L. Hua, K.C. Cheung, Stochastic orders of scalar products with applications, Insurance: Mathematics and Economics 42 (2008) 865-872].