Some Results On Ruin Probabilities In A Two-dimensional Risk Model

Ruin theory has always been a vital part of modern risk theory, and it is one of the popular topics of the risk theory. This paper is devoted to the development of the ruin theory for the two-dimensional risk model.Comparing with classical ruin meory in which the fact of one-dimensional has been considered, the multi- dimensional can reflect the business of the insurance company more accurately. Because of its complexity in mathematics, only few papers, such as Chan W.S.,Yang H.L.&Zhang L.Z.(2003)had discussed the topic under the model. Ruin theory under multi- dimensional risk models is very complex. Even in the two-dimensional case, the problem is challenging. Ruin probabilities with interest force about the classical risk theory are presented in the first part of the paper and extended to two-dimensional risk model. Based on the new risk model, we define three kinds of ruin probabilities and get their simple bounds.The risk process in which claim arrival is a normal renewal process that first was introduced by Sparre Andersen is another important generation for classical risk processes. Except of the progress of the mathematical technical, Andersen supposed the claims are infected, i.e. permitted the consideration of non-poisson claim arriving. Dickson&Hipp(1998) demonstrated risk process in which claim arriving is a normal renewal process by the method of deducing the classical risk model, especially Erlang (2) risk process. Under the situation of claim distribution is Phase-type, they obtained the explicit solutions for ruin probability and survival probability. Firstly, we introduce one-dimensional risk model in which claim inter-arrival times have an Erlang(2) distribution in the second part of this paper. Secondly, we approach explicit solutions for ruin probability in which claims have Phase-type distribution. At last, we construct a new two-dimensional risk model, define three kinds of ruin probabilities and get their simple bounds.