Bankruptcy And Dividends Under The Two Types Of Claims System

Risk theory is an important part of Actuarial Science,which is a general theory for quantita-tive analysis and prediction of risk,decision-making,control and management.Its research contents mainly include two aspects:one is the risk faced by the company,namely ruin theory;the other is the company's income,namely dividend strategy.The company's risk can be described by some actuarial quantities,such as the probability of ruin,the time of ruin,the surplus before ruin,the deficit at ruin and so on.Gerber and Shiu(1998)studied the joint distribution of the surplus before ruin,the time of riun and the deficit at ruin in the classical risk model,unified these production failures,and gave the expression of Gerber-Shiu function(the discounted penalty function)for the first time.Since then,most of the problems of studying ruin theory have been transformed into the problem of establishing Gerber-Shiu function to solve.In addition to risks,companies also care about their income.The most representative quantity of measuring company income is the total amount of the dividends before ruin,how to maximize the company's income has became a hot issue in risk theory research.Vierkott(2017)studies the optimal dividend strategy with penalty under the classical risk model.As a risk measurement standard,he considers the difference between the expected discounted dividend and the total amount of penalty.However,with the continuous expansion of the operation scale of insurance companies and the continuous development of new types of insurance,it is limited to use the risk model of a single type of insurance to describe the risk operation process.Therefore,this paper mainly extends the classic risk model to two types of claim systems,and studies the expected discounted penalty function and the optimal dividend strategy with penalty payment under the two types of compound Poisson claim system.Details as follows:In the chapter one,we mainly introduce the research background of risk theory.In the chapter two,we briefly introduce the classic risk model,and then the classic model is extended to two types of claim systems and two types of compound Poisson claim system models are established.we study the expected discounted penalty function and the moment of the time of ruin under two types of compound Poisson claims system.By using probability theory and Laplace transformation method,we can derive the Integro-differential equation satisfied by the ex-pected discounted penalty function,and by solving Gerber-Shiu function,we can obtain the specific expression of the expected discounted penalty function with the initial surplus u=0,and then obtain the moment of the time of ruin with the initial surplus u=0.When the claim amount of the two kinds of claims both obey the exponential distribution,the explicit solution of the expected discounted penalty function with the initial surplus u=0 is given.In the chapter three,we study the optimal dividend problem under two kinds of compound Poisson claim system.By using probability theory and martingale method,we can deduce the HJB equation satisfied by the value function V(x),and the prove the optimal dividend strate-gy is a boundary strategy.When the claim amount of two kinds of claims obey the exponential distribution,the case that the penalty function is an exponential function and a linear function is considered respectively.By solving the corresponding HJB equation,the explicit solution of the corresponding value function V(x)and the optimal boundary b*.when the claim amount of two kinds of claims both obey the exponential distribution,the case that the penalty function is an exponential function and a linear function are considered respectively,and by solving the corre-sponding HJB equation,we can obtain the explicit solution of the corresponding value function V(x)and the optimal dividend boundary b*.