Studies On The Risk Model With Interest

This dissertation is devoted to study the ruin problems in the risk model with stochastic rates of interest, the dividend problems in the risk model with constant interest and the ruin problems in the risk model with stochastic income of premium.From the classical risk model is introduced, many researcher generalize it for adapting to the actual running of the insurance company fearther. And the risk model with interest is one of generalizing on the classical risk model. In the traditional actuarial theory, interest factor is not considered generally. But when we consider a long-term risk process, we must consider the time value of money, which is interest problems. Recently some researcher at home or abroad begin to consider the interest factor. Yang.,Zhang(2001) abtain the joint distribution of surplus immediately before ruin and the deficit at ruin under interest force. Wu and Du(2002) discuss the renewal risk model in which the interest process is a constant. They obtain the probability of ruin, the distribution of surplus when ruin happen, series expansion and integal equation of the surplus immediately before ruin.Wu.,Wang.,Zhang(2005) derive the explicit expression for the joint distribution of three actuarial diagostics: the time of ruin,the surplus immediately before ruin and the deficit at ruin with constant interest force.Cai and Dickson (2002) study the Gerber-Shiu's expected discounted penalty function at ruin with stochastic rates of interest. Cai(2003) discuss the classical risk model with stochastic rates of interest and obtain the upper bounds of probability of ruin. On this base, Cai(2004) obtain the integro-differential equation for the Gerber-Shiu discounted penalty function. De Finetti(1957) put forward the optimal dividend problems firstly. He showed that, under the assumption that the surplus of the company is a discrete withsteps of size plus or minus one only, the optimal dividend-payment strategy is a barrier strategy. That is, any surplus above a centain level would be paid as dividends to the shareholders of the company. Buhlmann(1970) discuss the optimal dividend problem in the classical compound Poisson model. Gerber and Shiu(2004) study the maximal dividend problem in a barrier strategy with positive drift Brownian motion. Gerber and Shiu(2006) study the reflection and refraction problems on optimal dividends with positive drift Brownian motion. In the classical risk model, Gerber and Shiu(2006) discuss the dividend problem in a bounded rate, and they showed that the optimal dividend strategy is the barrier strategy. On this base, This dissertation study the dividend problem in a bounded rate in the classial risk model with intreest.Chapter 1 deals with the Erlang(2) risk model with stochastic rates oi interest, we consider the risk model for which the claim inter-arrival distribution is Erlang(2) and the stochastic interest process is a Levy process. We derive the integal equation, lower and upper bounds for ruin probability. When the interest process is assumed Brownian Motion or Brownian Motion with drift, we obtain the specific integral equation for ruin probability. Finally we discuss the penalty function, and give the integral equation and integro-differential equation for it.We obtained the following results : Theorem 1.2.1:Ruin probability satisfy the integal equation:Chapter 2 deals with the dividend problems in rate in the classical risk model with Interest. Based on Gerber and Shiu(2005),we discuss the dividend prooblems in rate in the classical risk model with interest.Under an optimal dividend strategy,an integral equation for the expectation of the aggregate dividents is obtained.When the initial surplus is greater than the barrier or is equal to the barrier, the explicit result of the aggregate dividends is obtainedChapter 3 deals with the premium income randomized risk model. We generalize the risk model of Gong Ri-zhao(2001),and randomize the premium income. Using martingale method,when the aggregate claims process and the premium income process are Poisson process,we discuss the probability of ruin.Then we discuss the Gerber-Shiu discounted penalty function,and obtain the integral equation and the Laplace transform for it.Finally.using the theory of random walks,we discuss the probability of ruin when the aggregate claims process and the premium income process are the same renewal process. We obtained the following results : Theorem 3.2.4: In this model, the probability of ruinWhen the aggregate claims process and the premium income process are the same renewal process :Theorem 3.4.4: When the distribution of claim amount Fy is exponential distribution with the parameter 6 , the probability of ruin is:Theorem 3.4.6: When the distribution of claim amount Fy is exponential distribution with the parameter 5 , the distribution of the premium income F\ is exponential distribution with the parameter a , the probability of ruin is:...