The Time Interval Of The Claim And The Risk Model Of The Interference With The Amount Of The Premium And The Amount Of The Claim

In the classical compound poisson risk model and the Sparre-Andersen risk model,one crucial assumption is that the interclaim times and the claim sizes are independent.Although such assumption simplifies the analysis of the ruin problems, this assumption might be inappropriate in practice. In this paper, we consider a perturbed risk process which the interclaim-time distribution and premium rate both depend on the size of the previous claim. A Lundberg-type equation is derived utilizing the Laplace transforms for the Gerber-Shiu function. The roots of the Lundberg equation are stadied. For exponential thresholds, the explicit expressions of the differential equations of the Gerber-Shiu function are given.In this paper, we model the dependence structure both interclaim times and premiums with claim sizes by the thresholds {Qi, i = 1, 2,...}. If Xjis larger than Qj, we classify the insured as Class 1, then the time until the next claim will follow an exponential distribution with mean(?)> 0 and charge premium at rate c1(> 0); If Xjis smaller than Qj, we classify the insured as Class 2, then the time until the next claim will follow an exponential distribution with mean(?)> 0, and charge premium at rate c2(> 0).we study a perturbed risk model with both interclaim times and premiums depending on claim sizes in the paper. The Gerber-Shiu functionin can be decomposed according to whether the ruin is caused by a claim or oscillation in the risk model. So we can consider the following four cases: ?i,w(u), i = 1, 2 is the Gerber-Shiu function where ruin is caused by a claim; ?i,d(u), i = 1, 2 is the Gerber-Shiu function where ruin is caused by a oscillation. Now we introduce a measure Pi(u, dy, dx), and we can conclude the expression of the Gerber-Shiu function by the measure, then the Lundberg’s equation of the Gerber-Shiu function can be obtained by the properties of the Translation operator.For δ > 0, δ = 0, we apply the Rouch(?)’s theorem on the contour C to discuss the roots of the Lunfberg’s equation.In chapter 3, for the exponential thresholds, we present the integral equation of the Gerber-Shiu function and apply the operator d/(du- ηi,j), i, j = 1, 2, to these equation,the differential equations of the Gerber-Shiu function are given.