Optimization Of Investment And Periodic Dividend In Compound Binomial Model

This paper mainly considers the optimization of investment and periodic dividend in compound binomial model under the condition of bounded dividend rate.The content of the first chapter is an introduction.This chapter first in-troduces the research background and significance of this paper,then states the history and current situation of the related research,and finally describes the research contents and structure of this paper.In the second chapter we give the basic model and hypothesis.Assuming that a certain proportion of risk investment and risk-free investment are made in each unit time,the premium is fixed,and the claim amounts are independent and identically distributed random variables.Insurance company periodically pays dividends.We will define the optimal value function and the corresponding optimal strategy for investments and dividend payments.The third chapter discusses the optimal value function and its transformation.Analysing the surplus process with investments and dividends,and using the full probability formula,we obtain the equation of the corresponding value function.This paper assumes that there is a certain upper bound of c₀ for dividend payments at any time.We prove that under any investment-dividend strategy,the value function has an upper bound.By transforming the value function,we define a new function W（u）in this chapter.The main content of Chapter 4 is to study the nature of the optimal strategy and the existence of the optimal W（u）.Under the condition of optimal W（u）existing,we prove the existence of optimal dividend and investment strategy.Then we prove that the optimal W（u）exists and is unique by the principle of contraction mapping.The main content of the fifth chapter is about the stochastic simulation of the optimal strategy.In order to solve the computational problem of optimal strategy,we propose a new method of stochastic simulation and provide theoretical proof for the effectiveness of the method.Chapter 6 gives an example for numerical calculation.We get the corre-sponding optimal dividend and investment strategy.We use a stochastic sim-ulation method to calculate the optimal strategy,which requires the computer to randomly generate limited sample points.Considering that the finiteness and randomness of sample points may affect the optimal strategy,firstly,we obtain the samples with different sample mean values;thereafter,samples with different population variances are obtained under the condition that the population mean is not changed.We will use these samples to calculate the corresponding optimal investment-dividend strategies,and compare and analyze the calculation results.In the end,we also discuss the optimal investment-dividend strategy with different dividend periods.