q-风险模型的伴随多维马尔可夫过程

Adjoint Multi-dimensional Markov Processes of a q-risk Model

q-风险模型是Q-风险模型的推广,而Q-风险模型又是经典风险模型的推广.在Q-风险模型中,索赔时刻是一个规则Q-过程的跳跃时刻,而索赔计数过程是马尔可夫到达过程(MAP)的计数过程,Q-风险模型被一个环境过程控制.环境过程是一个规则Q-过程,它取离散的实数值.实际问题中,环境过程可以取连续的实数值.因此,需要将环境过程Q-过程推广为q-过程,并进一步将Q-风险模型推广为q-风险模型.证明了q-风险模型的伴随2-维和3-维随机过程都是时齐马尔可夫过程,并求出了它们的初始分布和转移概率的显式表示式.作为q-风险模型的特殊情形,对Q-风险模型获得了相应的结论.

A q-risk model is a generalization of a Q-risk model,and a Q-risk model is a generalization of the classical risk model. In a Q-risk model the moments of claims are exactly the jumping moments of a regular Q-process and the counting process of claims is the counting process of a Markov arrival process （MAP）. The Q-risk model is controlled by an environment process. The environment process is a regular Q-process taking discrete real values. In practice an environment process may take general real values. So the environment process Q-process needs to be extend to a q-process. Furthermore a Q-risk model needs to be extend to a q-risk model. It is proved that the adjoint 2-dimensional and the 3-dimensional processes of a q-risk model are time-homogeneous Markov processes. The analytic expressions of their initial distrib and their transition probabilities are calculated. As a special case of a q-risk model the corresponding results of a Q- risk model are obtained.