The Compound Pascal Model With Dividends And Varying Premium Rate Under Random Environment

Risk theory has developed for more than one hundred years, and ruin theory is one of the coreissues in risk theory, studies of related ruin problems can give the early warning of risk during theinsurance companies’ running. Lundberg and Cramér first established the relationship between risktheory and stochastic process. Since then, the study of ruin theory has achieved great developments,and got a series of good results. However, with the development of market economy, the classical riskmodel can not describe the surplus of insurance company well. As the fund will not be left unused, thecompany will do various investments, so the factor of interest should not be ignored. The premiumthat policyholders pay to insurance company per unit time isn’t a constant, so it’s necessary to take thevarying premium rate into account. Furthermore, in joint stock insurance company, shareholders willrequest dividends when the surplus exceeds a certain level, also for some participating policies, thepolicyholders will ask the company to pay dividends to them.In this paper, for joint stock insurance company, we consider the case that the company willrandomly pay dividends to shareholders and policyholders respectively, and then establish thecompound pascal model that interest is modulated by a Markov chain and premium rate varies withsurplus level. First, we studied the homogeneous Markov property of the surplus process, and thengave the recursive formulas for finite time ruin probability, the integral equations for infinite time ruinprobability and joint distribution of the ruin time, the surplus before ruin and the deficit at ruin.Finally, we gave the numerical examples of finite time ruin probability.