On The Two Types Of Discrete-time Risk Models With Dividend Strategy

Discrete-time risk model with dividend strategy is a hot issue in risk theory.This thesis mainly studies two types of discrete-time risk models with dividend strategy.The first model is a discrete-time risk model with random premium and barrier dividend strategy.We introduce a dividend policy to the company,and once the insurer's surplus is reached b,the excess of the reserve over b will be paid to shareholders as dividends.The second model is a discrete-time risk model with general premium and threshold dividend strategy.Assuming that the insurance company will receive general premiums₁c at the beginning of each period,when the insurance company's surplus is reached b it will be paid?as dividends to shareholders immediately,and the insurance company will continue to operate with net premiums₂c at the next moment.The research content of this thesis is mainly divided into the following three parts.The first chapter introduces the research background of risk theory,including classic risk models and promotion forms,several basic forms of dividend strategy models and related research results,and finally briefly introduces the main research work in this thesis.The second chapter,we firstly establish a discrete-time risk model with random premium and barrier dividend strategy.Then use the difference equation theory to obtain the expected present value of dividend payments to satisfy the homogeneous difference equation.And use the Rouchétheorem to prove the distribution of the roots of the Lundberg equation inside or outside the unit circle and find the probability generating function of the fundamental solutions to obtain the defective renewal equation satisfied by the fundamental solutions of the difference equation.Finally,the analytical solution of the expected present value of dividend payments is given when the claim amount obeys the geometric distribution and the numerical analysis of the theoretical results is carried out.The third chapter,we firstly establish a discrete-time risk model with general premium and threshold dividend strategy,and the Gerber-Shiu expected discounted penalty function is divided into two parts according to the relationship between the initial surplus and the dividend barrier b.Then the probability generating function of the Gerber-Shiu expected discounted penalty function is obtained by using the difference equation theory,and the defective renewal equations satisfied by the fundamental solutions and the a particular solution of the Gerber-Shiu expected discounted penalty function are obtained by inverting the probability generating function.Finally,assuming that the claim amount obeys the second-order mixed geometric distribution,the analytical expression satisfied by the generating function of the time to ruin is obtained,and a numerical analysis is carried out.