The Problem Of Capital Injection And Dividend Payment In Compound Binomial Model

This article mainly explores the issue of capital injection and dividend distribution in the compound binomial model.Our research is based on the assumption that shareholders will try their best to bear all deficits.Its purpose is to enable insurance companies to operate continuously and so as to enable the welfare of our residents.The research on relevant issues under this assumption will fully demonstrate the social security function of insurance and has strong practical significance.The specific arrangements are as follow.Chapter 1 is the introduction of the article.This chapter mainly included: the research background and current situation,Key issues to be solved and innovation point,the main composition of the paper.Chapter 2 is preparatory knowledge.In this chapter,the basic knowledge involved in the paper is described in detail,and then the contraction mapping theory and related lemmas are introduced.Chapter 3 focus on the capital injection with or without capital injection fee and deficit penalty.For these two kinds of capital injection situations,we start with the time of 1 and the time when the claim first occurs respectively,and study the capital injection situation of the company,The equation satisfied by the expected present value of capital injection is derived.Then we prove the existence and uniqueness of the solution of the equation by the using compression mappings and the fixed point principle,and proved that the expected present value of capital injection is a reduction function of the initial surplus,the expected present value of capital injection without capital injection fee and penalty has nothing to do with the value of the capital injection boundary,but has something to do with value of the capital injection boundary minus the initial surplus,and other related theorems according to the nature of the equation.Finally,two examples are given for numerical calculation with MATLAB.Examples show that:(1)under the same conditions,the expected present value of capital injection calculated by using the equations established at two different update time points and the results are the same.At the same time,these numerical results also verify the correctness of the relevant theorems.Chapter 4 is to study the dividend problem with capital injection.This research on this problem not only has strong theoretical value,but also has practical application value.This chapter also studies from two different update time points,establishes two equations satisfied by the value function,and proves the existence and uniqueness of the solution of the two equations by using the contraction mapping and fixed points principle.At the end of this chapter,we also listed two examples to carry out numerical simulation through MATLAB,looking for the capital injection boundary and dividend boundary corresponding to the value function reaches the maximum.The last chapter summarizes the contents and results of this paper,and at the same time puts forward some problems that can be considered in future research.