The Gerber-Shiu Function Of The Bilateral Jump Risk Model Is Estimated Based On The Laguerre Series Expansion Method

Risk theory is a very important part in actuarial field,and the continuous development of the risk theory has also played a theoretically foundational role of the insurance company's price of insurance products,reinsurance proper arrangement,solvency effective management and the company bankruptcy accurate early warning.The expected discounted penalty function(Gerber-Shiu function)was proposed by Gerber and Shiu to study some common problems in risk theory.At present,Gerber-Shiu function has become a very important tool in the study of bankruptcy theory.However,we find that many characteristic quantities of the risk model are unknown to the insurance company in real life.Therefore,it is significantly important for us to study the expected discounted penalty function through the actual observed data in the market.In recent years,the Gerber-Shiu function were explored by more and more studies using the actual market observation data.The Gerber-Shiu function of a two-sided jumps risk model with constant and random premium income were mianly studied based on the actual observed data of earnings and claim process by using the Laguerre series expansion method in this paper.Firstly,we make a brief introduction for the Laguerre polynomial and the Laguerre series.And we use the defective renewal equation which satisfy the Gerber-Shiu function of the two-sided jumps model to verify the Gerber-Shiu function of the two-sided jumps model can be expanded by the Laguerre series.Secondly,we will use some research methods commonly used at the present stage to obtain the Laguerre series expansion coefficient of the Gerber-Shiu function of the studied risk model.The approximate value of the Gerber-Shiu function is obtained by truncating the Laguerre series,and the estimated value of the Gerber-Shiu function is obtained by using the actual observed data in the process of earnings and aggregate claims.Then,for the kind of two sided jumps risk model with constant and random premium income,we will analyze the error of the valuation results accordingly,and use some other tools such as L2-norm to get the convergence rate of the valuation,error of Gerber-Shiu function.In order to demonstrate the effectiveness of the proposed method,the Gerber-Shiu function of the studied model under different claim density functions is numerically simulated,and the effect of our estimation is verified by means of mean curve and integrated mean square error.Finally,it is found that the Laguerre series expansion method has a very small integrated mean square error,and as the observation time increases,the integrated mean square error becomes smaller and smaller.Therefore,we say that Laguerre series expansion method has the advantages of good approximation result and fast convergence rate.