Study On Two Kinds Of Generalized Risk Models

In this paper, the ruin problems of two types of risk models are mainly discussed. The first risk model is: the risk model with debit interest; The second type of risk model is: two classes of risk processes perturbed by diffusion.The risk model with debit interest which is based on the classical model, firstly proposed by Gerber, is a more practical risk model. Assume that the surplus of an insurer follows a compound Poisson surplus process perturbed by diffusion. When the surplus is below zero or the insurer is on debit, the insurer could borrow money at a debit interest rate to continue her business. Meanwhile the insurer will repay debt from her premium income. We assume that an insurer is allowed to continue her business with debt as long as her is at a reasonable level. However, when the debt or negative surplus is below some certain critical level, the insurer is no longer allowed to run his business. Absolute ruin occurs at this situation.In this paper, we will be more in-depth study of such model. Based on the original, adding the investment process.We assume that, whenever the surplus is negative or the company is on deficit, it could borrow an amount of money equal to the deficit at a debit interest forceδ' > 0; whenever the surplus is positive, the company could earn interest with forceδ> 0 for capital above a certain level b≥0, where b is the amount of capital the company retains as a liquid reserve. In this model, we study the expected discounted penalty functions. The integral and integro-differential equations for the expected discounted penalty functions are derived. For heavy-tailed claims, we obtain the asymptotic formula for the absolute ruin probability. In addition, when the claims are exponentially distributed, some explicit expressions for Gerber-Shiu functions are given. Finally, by a"renewal"argument, we obtain the explicit expression for the probability of the recovery when the claims are exponentially. The risk model with debit interest is more practical significance, the introduction of bank loans to the scene, more in line with the actual needs of today's society. The risk model is more complete, taking into account more realistic factors. This paper from the expected discounted penalty functions, for the absolute ruin of the other important studies related to the nature of the role of more counseling.At the same time, insurance companies operate with the continuous expansion of scale and new types of development, will inevitably lead to diversification, which leads to two Class of risk processes. In fact, unlike the classical risk model so idealized, insurance companies, the total claims will inevitably be affected by such factors as influence and interference, and this is Gerber proposed the classic risk model perturbed by diffusion, namely: the total insurance claim volume by the Wiener process disturbance, and this diffusion can be seen as the insurance company management or operation of the deviation of financial stability. This model greatly enhanced the original model for the description of reality. We assume that the two claim number processes are independent Poisson and generalized Erlang(n) processes, respectively. And W (t )is standard Wiener process which is the insurance companies express uncertainty of income or expenditure. Because it significantly strengthened the capacity of the original model description, in recent years gradually been theoretical circle and Services industry attention.In this paper, We will be the basis of the original model, the introduction of the dual realities of insurance factors a system of integro-differential equations satisfied by the survival probability is derived. Then we introduces a generalized Lundberg equation and discusses its roots. In addition, Laplace transforms ofφ'(0) are derived. Finally, Explicit results are derived when the claims size distributions, p and q , belong to K n family of distributions.