Studies On Ruin In The Risk Models With Constant Interest Rate

In the classical risk model, the number of claims from an insurance portfolio is assumed to follow a Poission process , the individual claim sizes are independent and identical random variables, and the premiums are described by a constant rate of income. In this kind of model, the clear expression for the ruin probability is given by Filip Lundberg and Cramer when the claim amount is exponentially distributed. Futhermore, they get the exponential upper bound by the aid of a renewal technique introduced by William Feller, which is also proved by Gerber(1973) using a martingale approach. The problem on the severity of ruin has recently received a remarkable attention. In Dufresne and Gerber(1988a,b),Gerber and Shiu(1997,1998),Gerber et al.(1987), Willmot and Lin(1997), and Yang and Zhang(2001a,b), the distributions of the ruin time, the surplus immediately prior to ruin and the deficit at ruin were considered.In the classical risk theory , it is often assumed that there is no investment income. However, as we know, a large portion of the surplus of the insurance companies comes from investment income. In recent years , the risk models with deterministic interest rate have been paid more attention. Sundt and Teugels(1995) considered the ultimate ruin probability in a compound Poisson model with a constant interest force, and they get its exact solution at the special case of exponential claim sizes. Yang(1999) considered a discrete time risk model with a constant interest force and both Lundberg-type inequality and non-exponential upper bounds for ruin probabilities were obtained by using martingale inequalities. Under the assumption of stochastic investment income , but a constant interest rate, a Lundberg-type inequality was obtained in Paulsen and Gjessing(1997).This thesis is devoted to a study of severity of ruin , upper bounds for ruin probabilityand retention levels for reinsurance in the risk models with a constant interest force. Concretely, four aspects of Works are considered:In the first part, we consider the expected value of a discounted function(u) associated with a constant force and the argument a in a Laplace transforms as a function of the initial surplus u .By using the techniques of renewal theory , we derive an integral equation for (u).We then find an exact solution for (0) .Therefore, the form of the solution for the expected value of a discounted function can be found.Hereafter, we consider the expected value of a discounted function connected with the joint and marginal distributions of the surplus immediately prior to ruin , the deficit at ruin and the ruin time, and obtain a relation among them, which generalise the results of Dickson(1992),Gerber and Shiu(1998) and Cai and Dickson(2002).Furthermore, we analysis of the properties of the joint and marginal moments of the surplus immediately prior to ruin , the deficit at ruin and the ruin time.Poisson-renewal risk model is such that the inter-occurrence time of guarantee slips' arrivals are independ and indentical exponential random variables , the number of claims follows an ordinary renewal process, and the number of guarantee slips is independ of that of claims . In the second part, two kinds of upper bounds in the model are obtained by martingale and recursive techniques respectively, and the numerical comparisons of upper bounds derived by each technique are presented.The upper bounds for the finite time ruin probability in Sparre Anderson risk model are derived.Sparre Anderson(1957) considered the ultimate ruin probability when claims occur as a general renewal process.Since then, non-Poissonian risk models have been drawn more attention.Malinovskii(1998) and Wang(2001) considered Laplace transform of the finite time ruin probability in Sparre Anderson risk model, and they give clear expressionfor the Laplace transforms when the individual claim sizes are exponentially and mixed exponentially distributed. In the third part, the upper bounds for the finite time ruin probability in Sparre Anderson risk model with a con...