The Research On The Risk Model With Stochastic Return

Since Segerdahl proposed stochastic return, as describe the uncertainty of returns more reasonable to practice, the risk model with stochastic return has become a more and more popular topic in risk theory. Then the model is possible to observe the current value of the surplus continuously. In practice, it is difficult to observate continuity. In this paper we introduced random observation times to the risk model with stochastic return. It may be more reasonable to practice. We mainly discuss ruin problems based on the risk model with stochastic return on investments and random observation periods. Through the Sine method to calculate the ruin probability. We obtain some results about ruin probability, the main ideas of these are as follows:In Chapter1, we presentation the background of the risk theory and its development. Then, we present the main work of this paper and the main result of my research.In Chapter2, we mainly introduce preliminary of the Sine function and some risk relevent models.In Chapter3, we discuss a risk model with stochastic return on investments and random observation periods. In practice, it may be more reasonable to assume that the balance of the book is only checked on a period basis. A integro-differential equation with certain boundary conditions are derived when the interclaim times have an exponential distribution. It is difficult to find the explicit solution, so we use the Sine methods to derive an approximate solution. Finally, study the effect of stochastic return and random observation periods on the ruin probability when the claim size distribution is exponential.In Chapter4, on the basis of the Chapter3, we consider the expected discounted penalty function a risk model with stochastic return on investments and random observation periods, when the interclaim times have a phase-type distribution. A integro-differential equation with certain boundary conditions is derived when the interclaim times have a exponential distribution.. It is difficult to find the explicit solution, so we use the Sinc methods to derive an approximate solution. Finally, study the effect of stochastic return and random observation periods on the ruin probability when the claim size distribution is exponential.In Chapter5, on the basis of the Chapter3, we introduce dividend policy: When the surplus below the level b1, no dividend; When the surplus is between the threshold b1and b2, premium income no longer goes into the surplus but as dividends to shareholders; When the surplus up the level b2, the b2-b1as the dividend to shareholders. We consider the expected discounted dividends and the expected discounted penalty function for a risk model with stochastic return on investments and random observation periods on investments and ran-dom observation periods. Integro-differential equations with certain boundary conditions are derived when the interclaim times have a exponential distribu-tion.