On The Compound Poisson Surplus Model With Liquid Reserves

In this thesis, we consider the compound Poisson risk model with debit inter-est and liquid reserves,and the compound Poisson risk model with double liquidreserves. In the compound Poisson risk model with debit interest and liquid re-serves,we assume that the company can borrow a sum of money which equal thedeficit when the surplus fall below zero while and begin returning the debit con-tinuously,and obtain the interest with the interest force r > 0 when the surplusabove the surplus level (?)≥0(which called the liquid level). In the compoundPoisson risk model with double liquid reserves,we assume that the company canobtain the interest with the larger interest force r₁ > 0 when the surplus between(?)1≥0 and (?)2≥0(0≤(?)1≤(?)2),and obtain the interest with the smaller inter-est force r₂ > 0(the missing part is used to dividends) when the surplus above thesurplus level (?)2. The expected discounted penalty (Gerber-Shiu ) function,whichis a very important function in insurance mathematics,contains the surplus beforeruin,the deficit at ruin,the ruin time and so on.The research of Gerber-Shiu functionis the core problem in the research of insurance mathematics. This thesis discussthe Gerber-Shiu function of these two models above,which make the research morereality. This thesis is divided into two chapters.In chapter 1, we introduce the classical compound Poisson risk model,the com-pound Poisson risk model with interest,the compound Poisson risk model with debitinterest,the compound Poisson risk model with liquid reserves,the compound Pois-son risk model with debit interest and liquid reserves,and the compound Poissonrisk model with double liquid reserves.Meanwhile we introduce the core problemin the actuarial literature, the expected discounted penalty function,and introducesome corresponding research developments.Then,we set a general framework for thisthesis.In chapter 2, we introduce the compound Poisson risk model with debit interestand liquid reserves. We derive the integro-di(?)erential equations and general solutionfor the expected discounted penalty (Gerber-Shiu) functions.Moreover,we derive theexplicit expressions of the Gerber-Shiu function when the claims are exponentiallydistributed.In chapter3, We introduce the compound Poisson risk model with double liquid reserves. We also derive the integro-di?erential equations and general solution forthe expected discounted penalty (Gerber-Shiu) functions.