In this paper, we consider the optimal insurance by minimizing the total loss of the insured under stop-loss order when the insured faces insurable risk as well as background risk. To depict the relationship between insurable risk and background risk, we apply stochastic orders proposed by Shaked and Shanthikumar (2007) to model positive or negative dependence. We derive the optimal contract forms under different dependence structures between insurable risk and background risk. We give the optimal insurance strategies when the sum of the insurable risk and background risk is stochastically decreasing or increasing in background risk. Furthermore, we consider the optimal insurance strategy under different dependence structures under two insurable risks and a single background risk.
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