An Inquiry Of The Properties Of Markov Modulated Risk Processes

Perfect results acquired upon the classic risk model, but the provision upon whichit is constructed is too restricted to adapt for widely applications. Crame′r-Lundbergmodel describe the arrive time of claims with the homogeneous poisson model, andassume the constant revenue rate. Some improvement is made on the distribution ofarriving time of claim and premium revenue rate and a difusion is added. As to thegood property of Le′vy process in describing the risk process, It is heavily studied today.Traditional risk models are used to study the ruin problems in one situation,though many good results are acquired, but the fixed parameters under one situationcannot well define the change of economic situations. The premium revenue rate issubstantially influenced by the economic environment, so is the value of the invest-ment portfolio. In the extreme situation, there is so great change of the intensity ofclaim and revenue rate. As we all known, business cycles exist, diferent kinds indexof economic fluctuate periodically with the time. Markov risk models under switchedenvironment describe the risk process in diferent situations with status change subjectto the transition matrix, is better suited for the risk under diferent economic environ-ment.Given that the change of measure of martingale or ruin occurs, Markov modulatedrisk model has remained good properties of PDMP. In the model of two states, whenthe claim has the exponential distribution, with the methods of laplace transformation,we acquire the ruin probability of closed form.At the end, we inquire the properties of two poisson risk models, acquire the fourorder ODE with the boundry condition.