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.     

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.     

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.     

A note on scale functions and the time value of ruin for Lévy insurance risk processes

We examine discounted penalties at ruin for surplus dynamics driven by a general spectrally negative Lévy process; the natural class of stochastic processes which contains many examples of risk processes which have already been considered in the existing literature. Following from the important contributions of [Zhou, X., 2005. On a classical risk model with a constant dividend barrier. North Am. Act. J. 95-108] we provide an explicit characterization of a generalized version of the Gerber-Shiu function in terms of scale functions, streamlining and extending results available in the literature.     

Lévy risk model with two-sided jumps and a barrier dividend strategy

In this paper, we consider a general Lévy risk model with two-sided jumps and a constant dividend barrier. We connect the ruin problem of the ex-dividend risk process with the first passage problem of the Lévy process reflected at its running maximum. We prove that if the positive jumps of the risk model form a compound Poisson process and the remaining part is a spectrally negative Lévy process with unbounded variation, the Laplace transform (as a function of the initial surplus) of the upward entrance time of the reflected (at the running infimum) Lévy process exhibits the smooth pasting property at the reflecting barrier. When the surplus process is described by a double exponential jump diffusion in the absence of dividend payment, we derive some explicit expressions for the Laplace transform of the ruin time, the distribution of the deficit at ruin, and the total expected discounted dividends. Numerical experiments concerning the optimal barrier strategy are performed and new empirical findings are presented.     

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.     

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.     

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.     

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．     

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 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.     

On the moments of the time to ruin in dependent Sparre Andersen models with emphasis on Coxian interclaim times

The structural properties of the moments of the time to ruin are studied in dependent Sparre Andersen models. The moments of the time to ruin may be viewed as generalized versions of the Gerber–Shiu function. It is shown that structural properties of the Gerber–Shiu function hold also for the moments of the time to ruin. In particular, the moments continue to satisfy defective renewal equations. These properties are discussed in detail in the model of Willmot and Woo (2012), which has Coxian interclaim times and arbitrary time-dependent claim sizes. Structural quantities needed to determine the moments of the time to ruin are specified under this model. Numerical examples illustrating the methodology are presented.     

On the threshold dividend strategy for a generalized jump-diffusion risk model

In this paper, we generalize the Cramér-Lundberg risk model perturbed by diffusion to incorporate jumps due to surplus fluctuation and to relax the positive loading condition. Assuming that the surplus process has exponential upward and arbitrary downward jumps, we analyze the expected discounted penalty (EDP) function of Gerber and Shiu (1998) under the threshold dividend strategy. An integral equation for the EDP function is derived using the Wiener-Hopf factorization. As a result, an explicit analytical expression is obtained for the EDP function by solving the integral equation. Finally, phase-type downward jumps are considered and a matrix representation of the EDP function is presented.     

Analysis of generalized Lévy white noise functionals

Employing the Segal-Bargmann transform (S-transform for abbreviation) of regular Lévy white noise functionals, we define and study the generalized Lévy white noise functionals by means of their functional representations acting on test functionals. The main results generalize (Gaussian) white noise analysis initiated by T. Hida to non-Gaussian cases. Thanks to the closed form of the S-transform of Lévy white noise functionals obtained in our previous paper, we are able to define and study the renormalization of products of Lévy white noises, multiplication operator by Lévy white noises, and the differential operators with respect to a Lévy white noise and their adjoint operators. In the courses of our investigation we also obtain a formula for the products of multiple Lévy-Itô stochastic integrals. As applications, we discuss the existence of Hitsuda-Skorokhod integral for Lévy processes, Kubo-Takenaka formula for Lévy processes, and Itô formula for generalized Lévy white noise functionals.     

De Finetti’s optimal dividends problem with an affine penalty function at ruin

In a Lévy insurance risk model, under the assumption that the tail of the Lévy measure is log-convex, we show that either a horizontal barrier strategy or the take-the-money-and-run strategy maximizes, among all admissible strategies, the dividend payments subject to an affine penalty function at ruin. As a key step for the proof, we prove that, under the aforementioned condition on the jump measure, the scale function of the spectrally negative Lévy process has a log-convex derivative.     

On the expected discounted penalty function for risk process with tax

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.     

The Gerber–Shiu discounted penalty functions for a risk model with two classes of claims

In this paper, we consider the ruin problems for a risk model involving two independent classes of insurance risks. We assume that the claim number processes are independent Poisson and generalized Erlang(n) processes, respectively. When the generalized Lundberg equation has distinct roots with positive real parts, both of the Gerber–Shiu discounted penalty functions with zero initial surplus and the Laplace transforms of the Gerber–Shiu discounted penalty functions are obtained. Finally, some explicit expressions for the Gerber–Shiu discounted penalty functions with positive initial surplus are given when the claim size distributions belong to the rational family.     

Quantum Lévy Laplacian and associated heat equation

As a non-commutative extension of the Lévy Laplacian for entire functions on a nuclear space, we define the quantum Lévy Laplacian acting on white noise operators. We solve a heat type equation associated with the quantum Lévy Laplacian and study its relation to the classical Lévy heat equation. The solution to the quantum Lévy heat equation is obtained also from a normal-ordered white noise differential equation involving the quadratic quantum white noise.