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1.
The Gerber-Shiu discounted penalty function in the risk process with phase-type interclaim times 总被引:1,自引:0,他引:1
In this paper, we consider the Gerber-Shiu discounted penalty function for the Sparre Anderson risk process in which the interclaim times have a phase-type distribution. By the Markov property of a joint process composed of the risk process and the underlying Markov process, we provide a new approach to prove the systems of integro-differential equations for the Gerber-Shiu functions. Closed form expressions for the Gerber-Shiu functions are obtained when the claim amount distribution is from the rational family. Finally we compute several numerical examples intended to illustrate the main results. 相似文献
2.
Jing-min He Rong Wu Hua-yue Zhang 《应用数学学报(英文版)》2008,24(1):117-128
In this paper we mainly study the ruin probability of a surplus process described by a piecewise deterministic Markov process (PDMP). An integro-differential equation for the ruin probability is derived. Under a certain assumption, it can be transformed into the ruin probability of a risk process whose premiums depend on the current reserves. Using the same argument as that in Asmussen and Nielsen, the ruin probability and its upper bounds are obtained. Finally, we give an analytic expression for ruin probability and its upper bounds when the claim-size is exponentially distributed. 相似文献
3.
We obtain the asymptotics of the Gerber-Shiu discounted penalty function in the classical Lundberg model. We cosider claims
from a class of subexponential distributions and find the asymptotics as the initial surplus x tends to infinity. The main term of the discounted penalty function ψ(x, δ) has different expressions in the cases where the interest rate δ > 0 and where δ = 0.
Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 598–605, October–December, 2006. 相似文献
4.
On the expected discounted penalty function at ruin of a surplus process with interest 总被引:3,自引:0,他引:3
In this paper, we study the expected value of a discounted penalty function at ruin of the classical surplus process modified by the inclusion of interest on the surplus. The ‘penalty’ is simply a function of the surplus immediately prior to ruin and the deficit at ruin. An integral equation for the expected value is derived, while the exact solution is given when the initial surplus is zero. Dickson’s [Insurance: Mathematics and Economics 11 (1992) 191] formulae for the distribution of the surplus immediately prior to ruin in the classical surplus process are generalised to our modified surplus process. 相似文献
5.
通过对带扰动项的Lévy风险过程的研究得到了其罚金折现期望(G-S)函数满足的更新方程,并给出了它的一个无穷级数表达式. 相似文献
6.
In this paper, we consider the Gerber-Shiu expected discounted penalty function for the perturbed compound Poisson risk process with constant force of interest. We decompose the Gerber-Shiu function into two parts: the expected discounted penalty at ruin that is caused by a claim and the expected discounted penalty at ruin due to oscillation. We derive the integral equations and the integro-differential equations for them. By solving the integro-differential equations we get some closed form expressions for the expected discounted penalty functions under certain assumptions. 相似文献
7.
In this paper, we consider a renewal risk model with stochastic premiums income. We assume that the premium number process and the claim number process are a Poisson process and a generalized Erlang (n) processes, respectively. When the individual stochastic premium sizes are exponentially distributed, the Laplace transform and a defective renewal equation for the Gerber-Shiu discounted penalty function are obtained. Furthermore, the discounted joint distribution of the surplus just before ruin and the deficit at ruin is given. When the claim size distributions belong to the rational family, the explicit expression of the Gerber-Shiu discounted penalty function is derived. Finally, a specific example is provided. 相似文献
8.
9.
Zhao Xia Ouyang Zisheng 《高校应用数学学报(英文版)》2005,20(3):289-296
In this article, the risk process perturbed by diffusion under interest force is considered, the continuity and twice continuous differentiability for Фδ(u,w) are discussed,the Feller expression and the integro-differential equation satisfied by Фδ (u ,w) are derived. Finally, the decomposition of Фδ(u,w) is discussed, and some properties of each decomposed part of Фδ(u,w) are obtained. The results can be reduced to some ones in Gerber and Landry's,Tsai and Willmot's, and Wang's works by letting parameter δ and (or) a be zero. 相似文献
10.
In this paper we consider the generalized Cramér-Lundberg risk model including tax payments. We investigate how tax payments affect the behavior of a Cramér-Lundberg surplus process by defining an expected discounted penalty function at ruin. We derive an explicit expression for this function by solving a differential equation. Consequently, the explicit formulas for the discounted probability density function of the surplus immediately before ruin and the discounted joint probability density function of the surplus immediately before ruin and the deficit at ruin are obtained. We also give explicit expressions for the function for exponential claims. 相似文献
11.
We consider that the reserve of an insurance company follows a renewal risk process with interest and dividend. For this risk process, we derive integral equations and exact infinite series expressions for the Gerber-Shiu discounted penalty function. Then we give lower and upper bounds for the ruin probability. Finally, we present exact expressions for the ruin probability in a special case of renewal risk processes. 相似文献
12.
In this paper, a risk model where claims arrive according to a Markovian arrival process (MAP) is considered. A generalization of the well-known Gerber-Shiu function is proposed by incorporating the maximum surplus level before ruin into the penalty function. For this wider class of penalty functions, we show that the generalized Gerber-Shiu function can be expressed in terms of the original Gerber-Shiu function (see e.g. [Gerber, Hans U., Shiu, Elias, S.W., 1998. On the time value of ruin. North American Actuarial Journal 2(1), 48-72]) and the Laplace transform of a first passage time which are both readily available. The generalized Gerber-Shiu function is also shown to be closely related to the original Gerber-Shiu function in the same MAP risk model subject to a dividend barrier strategy. The simplest case of a MAP risk model, namely the classical compound Poisson risk model, will be studied in more detail. In particular, the discounted joint density of the surplus prior to ruin, the deficit at ruin and the maximum surplus before ruin is obtained through analytic Laplace transform inversion of a specific generalized Gerber-Shiu function. Numerical illustrations are then examined. 相似文献
13.
Jie-hua XieWei Zou 《Journal of Computational and Applied Mathematics》2011,235(8):2392-2404
In this paper, we consider an extension to the compound Poisson risk model for which the occurrence of the claim may be delayed. Two kinds of dependent claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim and may be delayed with a certain probability. Both the expected discounted penalty functions with zero initial surplus and the Laplace transforms of the expected discounted penalty functions are obtained from an integro-differential equations system. We prove that the expected discounted penalty function satisfies a defective renewal equation. An exact representation for the solution of this equation is derived through an associated compound geometric distribution, and an analytic expression for this quantity is given for when the claim amounts from both classes are exponentially distributed. Moreover, the closed form expressions for the ruin probability and the distribution function of the surplus before ruin are obtained. We prove that the ruin probability for this risk model decreases as the probability of the delay of by-claims increases. Finally, numerical results are also provided to illustrate the applicability of our main result and the impact of the delay of by-claims on the expected discounted penalty functions. 相似文献
14.
In this paper, we extend the work of Mitric and Sendova (2010), which considered the absolute ruin problem in a risk model with debit and credit interest, to renewal and non-renewal structures. Our first results apply to MAP processes, which we later restrict to the Sparre Andersen renewal risk model with interclaim times that are generalized Erlang (n) distributed and claim amounts following a Matrix-Exponential (ME) distribution (see for e.g. Asmussen and O’Cinneide (1997)). Under this scenario, we present a general methodology to analyze the Gerber-Shiu discounted penalty function defined at absolute ruin, as a solution of high-order linear differential equations with non-constant coefficients. Closed-form solutions for some absolute ruin related quantities in the generalized Erlang (2) case complement the results obtained under the classical risk model by Mitric and Sendova (2010). 相似文献
15.
考虑信用风险模型的破产问题,研究Gerber-Shiu贴现罚函数,通过引进辅助模型,运用概率论的分析方法得到了其所满足的积分方程.相应地可以得到该模型下的破产概率、破产时刻前赢余和破产时刻赤字的联合分布及其边际分布,进一步完善了YangHailiang发表的相关问题的结果. 相似文献
16.
Guo-jingWang Chun-shengZhang RongWu 《应用数学学报(英文版)》2003,19(1):59-70
In this paper we consider the risk process that is described by a piecewise deterministic Markov processes(PDMP).We first present the construction of the risk process and them discuss some ruin problems for this new kind of risk model. 相似文献
17.
Gordon E. Willmot 《Insurance: Mathematics and Economics》2007,41(1):17-31
The defective renewal equation satisfied by the Gerber-Shiu discounted penalty function in the renewal risk model with arbitrary interclaim times is analyzed. The ladder height distribution is shown to be a mixture of residual lifetime claim severity distributions, which results in an invariance property satisfied by a large class of claim amount models. The class of exponential claim size distributions is considered, and the Laplace transform of the (discounted) defective density of the surplus immediately prior to ruin is obtained. The mixed Erlang claim size class is also examined. The simplified defective renewal equation which results when the penalty function only involves the deficit is used to obtain moments of the discounted deficit. 相似文献
18.
Yichun Chi 《Insurance: Mathematics and Economics》2010,46(2):385-396
In this paper, we extend the Cramér-Lundberg risk model perturbed by diffusion to incorporate the jumps of surplus investment return. Under the assumption that the jump of surplus investment return follows a compound Poisson process with Laplace distributed jump sizes, we obtain the explicit closed-form expression of the resulting Gerber-Shiu expected discounted penalty (EDP) function through the Wiener-Hopf factorization technique instead of the integro-differential equation approach. Especially, when the claim distribution is of Phase-type, the expression of the EDP function is simplified even further as a compact matrix-type form. Finally, the financial applications include pricing barrier option and perpetual American put option and determining the optimal capital structure of a firm with endogenous default. 相似文献
19.
In this paper,we consider a risk model in which each main claim may induce a delayed claim,called a by-claim.We assume that the time for the occurrence of a by-claim is random.We investigate the expected discounted penalty function,and derive the defective renewal equation satisfied by it.We obtain some explicit results when the main claim and the by-claim are both exponentially distributed,respectively.We also present some numerical illustrations. 相似文献
20.
The purpose of this paper is to consider the expected value of a discounted penalty due at ruin in the Erlang(2) risk process under constant interest force. An integro-differential equation satisfied by the expected value and a second-order differential equation for the Laplace transform of the expected value are derived. In addition, the paper will present the recursive algorithm for the joint distribution of the surplus immediately before ruin and the deficit at ruin. Finally, by the differential equation, the defective renewal equation and the explicit expression for the expected value are given in the interest-free case. 相似文献