Several Categories Including Positive And Negative Risks And Risk Models

In the category of insurance mathematics which is also called acturarial mathemetics, the ruin theory is the heart of the risk theory. During the recent decades years, Gerber and his colleague who have been working on the classical risk theory have got the fundamental theory of the classical risk theory. In recent years, Wang and his colleague who have been working on the negative risk theory have got the some conclusion of the negative risk theory which is correaponding to the classical risk theory.This thesis we add the negative risk theory to the classical risk theory. Then we will discuss the properties of the new model. This thesis is dividend into three sections according to content.Chapter one is the introduction. We first introduce the background and the development of the new model. Then we give the definition of the new model which is called the risk model containing a positive and a negative risks, it iswhere U₁(t) is the classical risk model, U₂(t) is the negative risk model, u—u₁ + u₂, c = c₁ + c₂, S(t) = S₁(t) + S₂(t).In Chapter two, we consider the Gerber-Shiu functionφ(u) of the new model,whereδ≥0, T means the ruin time, U(T-) means the surplus just before the ruin time, |U(T)| means the deficit at the ruin time,ω(x, y) is nonnegative function about x > 0 and y > 0. By considering the first claim of the new model, we derive a fundamental equation forφ(u). In the end, we consider the ruin probability of the the new risk model.In Chapter three, we add a diffusion process to the risk model containing a positive and a negative risks. The surplus at time t is nowU(t) = U₁(t) + U₂(t) +σW(t) = u + ct- S(t) +σW(t),t≥0In this chapter, we give two hypothesis of the claim process. Then we consider the ruin probability under each of the two hypothesis separately.