The bounds of premium and optimality of stop loss insurance under uncertain random environments

The potential loss of insured can be affected by many nondeterministic factors, in which uncertainty always coexists with randomness. Therefore, uncertain random variables are used to describe this phenomenon of simultaneous appearance of both uncertainty and randomness in potential loss. Based on that, the upper and lower bounds of premium with uncertain random loss are given, respectively. Moreover, a mathematical model of uncertain random optimal insurance problem is established and the stop loss insurance is proved to be the optimal insurance policy and the equation for calculating the optimal deductible is arrived. Some numerical examples are also given for illustration.