Asymptotic Behavior Of A Class Of Stochastic Poisson Process With Random Scale Factor

Large deviation theory is one of the important topics in applied probability theory. In which, the precise large deviation theory has been received more and more attentions of the scholars. It is generally known that those external events take place with small probability. But once they occur, the impact of these phenomenon is severe, even devastating, such as floods, earthquakes, etc. The claim caused by these extreme events will affect the insurance company, which brings large net losses. Generally, the net loss distributions are heavy-tailed. Many financial problems under the heavy-tailed distributions can be boiled down to the large deviation problems. So, it is very necessary to study the precise large deviation of the aggregate claims of heavy-tailed. In this paper, we present a new class of Poisson process,whose externa’ intensity process has stochastic scaled factors. We study its precise large deviations under some conditions. Our study is divided into the following three parts:In chapter 1, we first introduce the concepts of the light-tailed distribution and several types of heavy-tailed distributions. Then, we introduce the development process of the precise large deviation theory of heavy-tailed distribution and the main conclusions. Finally, we draw the main research contents of this paper.In chapter 2, we describe the large deviation conclusions of two kinds of doubly stochastic processes. One is the common Cox process, it mainly generalizes the intensity of general Poisson process. In the description of the Cox process, we first discuss its asymptotic properties and get the asymptotic distribution of it. Secondly, this paper introduces the results of the precise large deviation of the Cox process under the conditions of the extended regular tail. In Theorem 2.3, we extend the Theorem 2.2 from extended regular tail condition to a larger class of heavy tailed distribution C class condition. The other is the doubly stochastic Poisson process, whose counting process is the general renewal counting process. Its difference from Cox process lies in the internal counting process. We give the conclusion of precise large deviation in the C class.The chapter 3 is the main content of this paper. We study large deviation of the doubly stochastic Poisson process with random scaled factors in the external intensity process. We give the expectation of the doubly stochastic Poisson process N(t) and the random sums S(t) of non-negative random variables series{Xi,i≥1} respectively. In the first place, we assume that the ratio of N(t) and its expectation converges to one in probability. At the same time, it is assumed that the exponential moment of N(t) tends to zero. Then, the asymptotic results are obtained. That is to say, the tail probability of difference between S(t) and E(S(t)) can be estimated by the expectation of the doubly stochastic Poisson process and the tail probability of Xj. Secondly, we give a new condition of external intensity process on doubly stochastic Poisson process and prove the corresponding large deviation under this condition.