Absolute ruin in the compound Poisson model with credit and debit interests and liquid reserves

In this paper, we study the absolute ruin probability in the compound Poisson model with credit and debit interests and liquid reserves. At first, we derive a system of integro‐differential equations with certain boundary conditions for the Gerber–Shiu function. Then, applying these results, we obtain asymptotical formula of the absolute ruin probability for subexponentially claims. Furthermore, when the claims are exponentially distributed, we obtain the explicit expressions for the Gerber–Shiu function and the exact solution for the absolute ruin probability. Finally, we discuss the absolute ruin probability by using the Gerber–Shiu function when debit interest is varying. In the case of exponential individual claim, we give the explicit expressions for the Gerber–Shiu function. Copyright © 2012 John Wiley & Sons, Ltd.     

复合Poisson-geometric风险模型下第n次索赔时的破产概率研究

在复合Poisson-geometric风险模型下,通过构造一个特殊的Gerber-Shiu函数,推导出此风险模型下Gerber-Shiu函数满足的更新方程,破产时刻和直到破产时的索赔次数的联合密度函数,得到了第n次索赔时的破产概率的数学表达式．     

On the compound Poisson risk model with dependence based on a generalized Farlie–Gumbel–Morgenstern copula

In this paper we consider an extension to the classical compound Poisson risk model in which we introduce a dependence structure between the claim amounts and the interclaim time. This structure is embedded via a generalized Farlie–Gumbel–Morgenstern copula. In this framework, we derive the Laplace transform of the Gerber–Shiu discounted penalty function. An explicit expression for the Laplace transform of the time of ruin is given for exponential claim sizes.     

On a multi-threshold compound Poisson process perturbed by diffusion

We consider a multi-layer compound Poisson surplus process perturbed by diffusion and examine the behaviour of the Gerber–Shiu discounted penalty function. We derive the general solution to a certain second order integro-differential equation. This permits us to provide explicit expressions for the Gerber–Shiu function depending on the current surplus level. The advantage of our proposed approach is that if the diffusion term converges to zero, the above-mentioned explicit expressions converge to those under the classical compound Poisson model, provided that the same initial conditions apply. This is subsequently illustrated by an extended example related to the probability of ultimate ruin.     

Erlang(2)过程的风险分析与美式看跌期权

在经典的风险理论中涉及到的索赔风险是服从复合Poission过程的, 与之不同, 我们考虑Erlang(2)风险过程\bd Erlang(2)分布往往见诸于控制理论中, 这里它作为索赔发生间隔时间的分布被引入了\bd 本文中, 我们介绍一个与破产时刻、破产前时刻的盈余以及破产时刻赤字有关的辅助函数$\phi(\cdot)$, 函数中涉及的这三个变量对风险模型的研究都是最基本也是最重要的\bdWillmot and Lin (1999)曾在古典连续时间风险模型之中研讨过这一函数\bd受Gerber and Shi(1997)及Willmot and Lin (2000)在古典模型下的研究过程的启发, 本文的一个重要结果就是找到破产前时刻的盈余以及破产时刻赤字的联合分布密度函数\bd 更得益于Gerber and Landry (1998)及Gerber and Shiu (1999)的思想, 我们应用以上的结果去寻求基础资产服从一定风险资产价格过程的美式看跌期权最优交易策略.     

On a generalization of the Gerber-Shiu function to path-dependent penalties

The Expected Discounted Penalty Function (EDPF) was introduced in a series of now classical papers ( [Gerber and Shiu, 1997], [Gerber and Shiu, 1998a] and [Gerber and Shiu, 1998b]). Motivated by applications in option pricing and risk management, and inspired by recent developments in fluctuation theory for Lévy processes, we study an extended definition of the expected discounted penalty function that takes into account a new ruin-related random variable. In addition to the surplus before ruin and deficit at ruin, we extend the EDPF to include the surplus at the last minimum before ruin. We provide an expression for the generalized EDPF in terms of convolutions in a setting involving a subordinator and a spectrally negative Lévy process. Some expressions for the classical EDPF are recovered as special cases of the generalized EDPF.     

Pricing perpetual American catastrophe put options: A penalty function approach

The expected discounted penalty function proposed in the seminal paper by Gerber and Shiu [Gerber, H.U., Shiu, E.S.W., 1998. On the time value of ruin. North Amer. Actuarial J. 2 (1), 48-78] has been widely used to analyze the joint distribution of the time of ruin, the surplus immediately before ruin and the deficit at ruin, and the related quantities in ruin theory. However, few of its applications can be found beyond except that Gerber and Landry [Gerber, H.U., Landry, B., 1998. On the discount penalty at ruin in a jump-diffusion and the perpetual put option. Insurance: Math. Econ. 22, 263-276] explored its use for the pricing of perpetual American put options. In this paper, we further explore the use of the expected discounted penalty function and mathematical tools developed for the function to evaluate perpetual American catastrophe equity put options. We obtain the analytical expression for the price of perpetual American catastrophe equity put options and conduct a numerical implementation for a wide range of parameter values. We show that the use of the expected discounted penalty function enables us to evaluate the perpetual American catastrophe equity put option with minimal numerical work.     

Pricing credit default swaps with a random recovery rate by a double inverse Fourier transform

We evaluate the par spread for a single-name credit default swap with a random recovery rate. It is carried out under the framework of a structural default model in which the asset-value process is of infinite activity but finite variation. The recovery rate is assumed to depend on the undershoot of the asset value below the default threshold when default occurs. The key part is to evaluate a generalized expected discounted penalty function, which is a special case of the so-called Gerber–Shiu function in actuarial ruin theory. We first obtain its double Laplace transform in time and in spatial variable, and then implement a numerical Fourier inversion integration. Numerical experiments show that our algorithm gives accurate results within reasonable time and different shapes of spread curve can be obtained.     

具有分红、流动储备金和利率的风险模型的绝对破产

建立了阈值分红策略下具有流动储备金、投资利率和贷款利率的复合泊松风险模型.利用全概率公式和泰勒展式,推导出了该模型的Gerber-Shiu函数和绝对破产时刻的累积分红现值期望满足的积分-微分方程及边界条件,借助Volterra方程,给出了Gerber-Shiu函数的解析表达式.     

Optimal reinsurance for Gerber–Shiu functions in the Cramér–Lundberg model

Complementing existing results on minimal ruin probabilities, we minimize expected discounted penalty functions (or Gerber–Shiu functions) in a Cramér–Lundberg model by choosing optimal reinsurance. Reinsurance strategies are modeled as time dependent control functions, which lead to a setting from the theory of optimal stochastic control and ultimately to the problem’s Hamilton–Jacobi–Bellman equation. We show existence and uniqueness of the solution found by this method and provide numerical examples involving light and heavy tailed claims and also give a remark on the asymptotics.     

On a class of stochastic models with two-sided jumps

In this paper a stochastic process involving two-sided jumps and a continuous downward drift is studied. In the context of ruin theory, the model can be interpreted as the surplus process of a business enterprise which is subject to constant expense rate over time along with random gains and losses. On the other hand, such a stochastic process can also be viewed as a queueing system with instantaneous work removals (or negative customers). The key quantity of our interest pertaining to the above model is (a variant of) the Gerber–Shiu expected discounted penalty function (Gerber and Shiu in N. Am. Actuar. J. 2(1):48–72, 1998) from ruin theory context. With the distributions of the jump sizes and their inter-arrival times left arbitrary, the general structure of the Gerber–Shiu function is studied via an underlying ladder height structure and the use of defective renewal equations. The components involved in the defective renewal equations are explicitly identified when the upward jumps follow a combination of exponentials. Applications of the Gerber–Shiu function are illustrated in finding (i) the Laplace transforms of the time of ruin, the time of recovery and the duration of first negative surplus in the ruin context; (ii) the joint Laplace transform of the busy period and the subsequent idle period in the queueing context; and (iii) the expected total discounted reward for a continuous payment stream payable during idle periods in a queue.     

Optimal dividends with incomplete information in the dual model

In [Gerber, H.U., Shiu, E.S.W., Smith, N., 2008. Methods for estimating the optimal dividend barrier and the probability of ruin. Insurance: Math. Econ. 42 (1), 243-254], methods were analyzed for estimating the optimal dividend barrier (in the sense of de Finetti). In particular, De Vylder approximations and diffusion approximations are discussed. These methods are useful when only the first few moments of the claim amount distribution are known.The purpose of this paper is to examine these and other methods (such as the gamma approximations and the gamproc approximations) in the dual model, see [Avanzi, B., Gerber, H.U., Shiu, E.S., 2007. Optimal dividends in the dual model. Insurance: Math. Econ. 41 (1), 111-123]. The dual model is obtained if the roles of premiums and claims are exchanged. In other words, the company has random gains, which constitute a compound Poisson process, and expenses occur continuously at a constant rate. The approximations can easily be implemented, and their accuracy is surprisingly good. Several numerical illustrations enhance the paper.     

A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model

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.     

Extended Gerber–Shiu functions in a risk model with interest

We consider a compound Poisson risk model with interest. The Gerber–Shiu discounted penalty function is modified with an additional penalty for reaching a level above the initial capital. We show that the problem can be split into two independent problems; an original Gerber–Shiu function and a first passage problem. We also consider the case of negative interest. Finally, we apply the results to a model considered by Embrechts and Schmidli (1994).     

On the dual risk model with tax payments

In this paper, we study the dual risk process in ruin theory (see e.g. Cramér, H. 1955. Collective Risk Theory: A Survey of the Theory from the Point of View of the Theory of Stochastic Processes. Ab Nordiska Bokhandeln, Stockholm, Takacs, L. 1967. Combinatorial methods in the Theory of Stochastic Processes. Wiley, New York and Avanzi, B., Gerber, H.U., Shiu, E.S.W., 2007. Optimal dividends in the dual model. Insurance: Math. Econom. 41, 111–123) in the presence of tax payments according to a loss-carry forward system. For arbitrary inter-innovation time distributions and exponentially distributed innovation sizes, an expression for the ruin probability with tax is obtained in terms of the ruin probability without taxation. Furthermore, expressions for the Laplace transform of the time to ruin and arbitrary moments of discounted tax payments in terms of passage times of the risk process are determined. Under the assumption that the inter-innovation times are (mixtures of) exponentials, explicit expressions are obtained. Finally, we determine the critical surplus level at which it is optimal for the tax authority to start collecting tax payments.     

Estimating the Gerber–Shiu function in the perturbed compound Poisson model by Laguerre series expansion

In this paper, we study the statistical estimation of the Gerber–Shiu function in the compound Poisson risk model perturbed by diffusion. This problem has been solved in [32] by the Fourier–Sinc series expansion method. Different from [32], we use the Laguerre series to expand the Gerber–Shiu function and propose a relevant estimator. The estimator is easily computed and has fast convergence rate. Various simulation studies are presented to confirm that the estimator performs well when the sample size is finite.     

The Compound Poisson Surplus Model with Interest and Liquid Reserves: Analysis of the Gerber–Shiu Discounted Penalty Function

We modify the compound Poisson surplus model for an insurer by including liquid reserves and interest on the surplus. When the surplus of an insurer is below a fixed level, the surplus is kept as liquid reserves, which do not earn interest. When the surplus attains the level, the excess of the surplus over the level will receive interest at a constant rate. If the level goes to infinity, the modified model is reduced to the classical compound Poisson risk model. If the level is set to zero, the modified model becomes the compound Poisson risk model with interest. We study ruin probability and other quantities related to ruin in the modified compound Poisson surplus model by the Gerber–Shiu function and discuss the impact of interest and liquid reserves on the ruin probability, the deficit at ruin, and other ruin quantities. First, we derive a system of integro-differential equations for the Gerber–Shiu function. By solving the system of equations, we obtain the general solution for the Gerber–Shiu function. Then, we give the exact solutions for the Gerber–Shiu function when the initial surplus is equal to the liquid reserve level or equal to zero. These solutions are the key to the exact solution for the Gerber–Shiu function in general cases. As applications, we derive the exact solution for the zero discounted Gerber–Shiu function when claim sizes are exponentially distributed and the exact solution for the ruin probability when claim sizes have Erlang(2) distributions. Finally, we use numerical examples to illustrate the impact of interest and liquid reserves on the ruin probability.      

On a Risk Model with Surplus-dependent Premium and Tax Rates

In this paper, the compound Poisson risk model with surplus-dependent premium rate is analyzed in the taxation system proposed by Albrecher and Hipp (Bl?tter der DGVFM 28(1):13–28, 2007). In the compound Poisson risk model, Albrecher and Hipp (Bl?tter der DGVFM 28(1):13–28, 2007) showed that a simple relationship between the ruin probabilities in the risk model with and without tax exists. This so-called tax identity was later generalized to a surplus-dependent tax rate by Albrecher et al. (Insur Math Econ 44(2):304–306, 2009). The present paper further generalizes these results to the Gerber–Shiu function with a generalized penalty function involving the maximum surplus prior to ruin. We show that this generalized Gerber–Shiu function in the risk model with tax is closely related to the ‘original’ Gerber–Shiu function in the risk model without tax defined in a dividend barrier framework. The moments of the discounted tax payments before ruin and the optimal threshold level for the tax authority to start collecting tax payments are also examined.