Study On Some Kinds Of Markovian Risk Model

Risk theory, one of the important branches of applied probability, plays a key rolein the area of insurance mathematics. As the risk theory in all aspects of theapplication becomes increasingly widespread, the in-depth study of models becomesmore and more important. The classical risk model1is not diffused by the externalfactors, but with the risk theory development and the needs of real life, people beganto consider the impact of external factors on the model, such as interest rate, dividendsand disturbance27. In this paper, we study the risk theory with interest rate, dividendsand disturbance in the Markovian environment. The main ideas and contributions ofthis paper are as follow:First, basing on Markov dependent model which was proposed by Janssen andReinhard8, then we add a constant interest23, we can establish a kind of risk modelsin Markovian environment. We got the integro-differential equations with boundaryconditions. Also, an example where the intensity process is a two-state Markov processand the claims are exponentially distributed is given.Second, Define the new dividend policy, we derive integro-differential equationsfor the content of the expected discounted dividend function Vi u;b, the content ofthe moment generating function M i(u, b;b)and the Gerber-Shiu function. Then wegot the integro-differential equations for the n order moment, so we can analyze thehigher moments of the present value of all dividend payments prior to ruin.Third, on the basis of the above model, we study a Markov-dependent risk modelwith perturbed, and consider investing the insurance company’s asset free risk. Usingthe Markovian property and the knowledge of stochastic differential equation, weobtain the integro-differential equations of the expected discounted total dividendpayments, its moment generating function and Gerber-Shiu function.Fourth, on the basis of the model with perturbed, we consider the dividend policyand the loan interest. Then we derive integro-differential equations for the content of the expected discounted dividend function Vi u;b, the content of the momentgenerating function M i(u, b;b)and the Gerber-Shiu function. Then we got theintegro-differential equations for the n order moment.