The Application Of Markov Chain In The Risk Theory And Reverse Mortgage

The Markov process, which named after Russia mathematician A.A. Markov, is a stochastic process. It has been widely used not only in other branches of mathematic , engineering technique, but also in the social science, for example: economics, insurance, financial risk, risk manage theory and technique. In this paper, I did some researches in the application of the Markov chain method of risk theory model and reverse mortgage model.The Risk ModelThe classical risk model has almost been perfected. In the continuous time model, we always use the Gerber-Shiu (Expected Discounted) Penalty Function with excellent behaves. Compared with the continuous time model, the discrete time model has some characteres itself , so the method above seldom been used in this case . In this paper, we researched two risk theory models with Markov chain method:1. Dual Poisson model with constant interest rate (chapter2), namely both the charge and claim of premium in the surplus process are compound Poisson processes, and the surplus produce interest,δ(≥0) is the interest rate density.2. The discrete time risk model with stochastic interest rate (chapter3).Based on the surplus process has the Markov property, we can use the method of Markov chain in the research of risk theory .we get some common risk quantities with explicit form by using the Markov Transition Matrix. Furthermore, we get the approximate calculate formula, the error estimate expression, an upper bound and a lower bound of the ruin probability. Just make use of this kind to get formula to computer risk quantities , the mount of calculation is very great, and along with increasing of the calculate precision,the quantity will be more, maybe this is one of the reasons that seldom touch on this method in the former research. We now can solve the question along with the rapid development of computer technique. Because of the Markov Transition Matrix is the perfect method to deal with the discrete time model, we think this method has widely applicational foreground to express the ruin quantity. Reverse Mortgage ModelWith the development of the economic, the increase of the people's live level, the progresses of the medical technique and the opening of the people's idea, the increase of obvious average people life , oldize society is coming. So the reverse mortgage, a operational way that can make the social security marketize, which adopts the financial instrument to collocate life-time source, has become one of the popularity hot problem. The reverse mortgage can make it fact that turn the housing fixed assets into cash flow. In chapter1,we expatiated the concept of the reverse mortgage, the domestic and overseas researchful actuality and the necessary to china. In chapter 4, we studied the paying in month problem of the reverse mortgage by using the theory and method of probability in the strict mathematics angle, this is so-called the reverse mortgage pricing problem. In this chapter, we also studied the reverse mortgage model that the interest rate obeying to the Markov chain, founded a commonly reverse mortgage model-which interest rate obey the Markov chain, get the pricing equation of the reverse mortgage model and obtained some explicit formulas in special situations.Those dates in this paper are get from programming and computering by S-Plus statistical software.