The phase-type risk model perturbed by diffusion under a threshold dividend strategy

This paper considers a perturbed renewal risk process in which the inter-claim times have a phase-type distribution under a threshold dividend strategy. Integro-differential equations with certain boundary conditions for the moment-generating function and the mth moment of the present value of all dividends until ruin are derived. Explicit expressions for the expectation of the present value of all dividends until ruin are obtained when the claim amount distribution is from the rational family. Finally, we present an example.     

A Joint Density Function in Phase-Type (2) Risk Model

In this paper, we consider a Gerber-Shiu discounted penalty function in Sparre Andersen risk process in which claim inter-arrival times have a phase-type (2) distribution, a distribution with a density satisfying a second order linear differential equation. By conditioning on the time and the amount of the first claim, we derive a Laplace transform of the Gerber-Shiu discounted penalty function, and then we consider the joint density function of the surplus prior to ruin and the deficit at ruin and some ruin related problems. Finally, we give a numerical example to illustrate the application of the results.     

Two parallel insurance lines with simultaneous arrivals and risks correlated with inter-arrival times

We investigate an insurance risk model that consists of two reserves which receive income at fixed rates. Claims are being requested at random epochs from each reserve and the interclaim times are generally distributed. The two reserves are coupled in the sense that at a claim arrival epoch, claims are being requested from both reserves and the amounts requested are correlated. In addition, the claim amounts are correlated with the time elapsed since the previous claim arrival.We focus on the probability that this bivariate reserve process survives indefinitely. The infinite-horizon survival problem is shown to be related to the problem of determining the equilibrium distribution of a random walk with vector-valued increments with ‘reflecting’ boundary. This reflected random walk is actually the waiting time process in a queueing system dual to the bivariate ruin process.Under assumptions on the arrival process and the claim amounts, and using Wiener–Hopf factorization with one parameter, we explicitly determine the Laplace–Stieltjes transform of the survival function, c.q., the two-dimensional equilibrium waiting time distribution.Finally, the bivariate transforms are evaluated for some examples, including for proportional reinsurance, and the bivariate ruin functions are numerically calculated using an efficient inversion scheme.     

Optimal dynamic excess-of-loss reinsurance and multidimensional portfolio selection

In this paper, the surplus process of the insurance company is described by a Brownian motion with drift. In addition, the insurer is allowed to invest in a risk-free asset and n risky assets and purchase excess-of-loss reinsurance. Under short-selling prohibition, we consider two optimization problems: the problem of maximizing the expected exponential utility of terminal wealth and the problem of minimizing the probability of ruin. We first show that the excess-of-loss reinsurance strategy is always better than the proportional reinsurance under two objective functions. Then, by solving the corresponding Hamilton-Jacobi-Bellman equations, the closed-form solutions of their optimal value functions and the corresponding optimal strategies are obtained. In particular, when there is no risky-free interest rate, the results indicate that the optimal strategies, under maximizing the expected exponential utility and minimizing the probability of ruin, are equivalent for some special parameter. This validates Ferguson’s longstanding conjecture about the relation between the two problems.     

Explicit ruin formulas for models with dependence among risks

We show that a simple mixing idea allows one to establish a number of explicit formulas for ruin probabilities and related quantities in collective risk models with dependence among claim sizes and among claim inter-occurrence times. Examples include compound Poisson risk models with completely monotone marginal claim size distributions that are dependent according to Archimedean survival copulas as well as renewal risk models with dependent inter-occurrence times.     

带干扰phaes-type风险模型中的最大盈余及相关分布

本文考虑了索赔时间间距为phase-type分布时带干扰更新风险模型中的破产前最大盈余、破产后赤字的分布,建立了相应的积分-微分方程.最后,讨论了当索赔时间间距为Erlang(2)分布且索赔量满足指数分布时的特殊情形.     

The expectation of aggregate discounted dividends for a Sparre Anderson risk process perturbed by diffusion

In this paper, we study the expectation of aggregate dividends until ruin for a Sparre Andersen risk process perturbed by diffusion under a threshold strategy, in which claim waiting times have a common generalized Erlang(n) distribution. For this strategy, we assume that if the surplus is above certain threshold level before ruin, dividends are continuously paid at a constant rate that does not exceed the premium rate, and if not, no dividends are paid. We obtain some integro-differential equations satisfied by the expected discounted dividends, and further its renewal equations. Finally, applying these results to the Erlang(2) risk model perturbed by diffusion, where claims have a common exponential distributions, we give some explicit expressions and numerical analysis. Copyright © 2007 John Wiley & Sons, Ltd.     

Ruin Probability in a Correlated Aggregate Claims Model with Common Poisson Shocks: Application to Reinsurance

This paper considers a correlated aggregate claims model with common Poisson shocks, which allows for dependence in n (n ≥ 2) classes of business across m (m ≥ 1) different types of stochastic events. The dependence structure between different claim numbers is connected with the thinning procedure. Under combination of quota-share and excess of loss reinsurance arrangements, we examine the properties of the proposed risk model. An upper bound for the ruin probability determined by the adjustment coefficient is established through martingale approach. We reduce the problem of optimal reinsurance strategy for maximizing the insurer’s adjustment coefficient and illustrate the results by numerical examples.     

The maximum surplus before ruin for two classes of perturbed risk model

In this paper, we consider the distribution of the maximum surplus before ruin in a perturbed risk model with two independent classes of risks, in which both of the two inter-claim times have phase-type distributions. We obtain the integro-differential equations for the distribution of the maximum surplus before ruin. Explicit expressions are derived if the two classes claim amount distributions both belong to the rational family.     

The perturbed compound Poisson risk model with constant interest and a threshold dividend strategy

In this paper, we consider the compound Poisson risk model perturbed by diffusion with constant interest and a threshold dividend strategy. Integro-differential equations with certain boundary conditions for the moment-generation function and the nth moment of the present value of all dividends until ruin are derived. We also derive integro-differential equations with boundary conditions for the Gerber-Shiu functions. The special case that the claim size distribution is exponential is considered in some detail.     

The perturbed Sparre Andersen model with a threshold dividend strategy

In this paper, we consider a Sparre Andersen model perturbed by diffusion with generalized Erlang(n)-distributed inter-claim times and a threshold dividend strategy. Integro-differential equations with certain boundary conditions for the moment-generation function and the mth moment of the present value of all dividends until ruin are derived. We also derive integro-differential equations with boundary conditions for the Gerber–Shiu functions. The special case where the inter-claim times are Erlang(2) distributed and the claim size distribution is exponential is considered in some details.     

一类索赔相关且带干扰的风险模型

该文讨论了索赔时间间隔与索赔量相关且带干扰的风险模型. 借助拉普拉斯变换研究了此模型的破产时刻、破产前瞬间盈余及破产时赤字三者的联合分布,得到了此联合分布拉普拉斯变换的一个分析表达式并讨论了当初始盈余值趋于无穷大时,此联合分布的渐近表达式.     

The maximum surplus before ruin and related problems in a jump-diffusion renewal risk process

In this paper, we investigate a Sparre Andersen risk model perturbed by diffusion with phase-type inter-claim times. We mainly study the distribution of maximum surplus prior to ruin. A matrix form of integro-differential equation for this quantity is derived, and its solution can be expressed as a linear combination of particular solutions of the corresponding homogeneous integro-differential equations. By using the divided differences technique and nonnegative real part roots of Lundberg’s equation, the explicit Laplace transforms of particular solutions are obtained. Specially, we can deduce closed-form results as long as the individual claim size is rationally distributed. We also give a concise matrix expression for the expected discounted dividend payments under a barrier dividend strategy. Finally, we give some examples to present our main results.     

Optimal proportional reinsurance and investment with minimum probability of ruin

The paper concerns a problem of optimal reinsurance and investment in order to minimizing the probability of ruin. In the whole paper, the cedent’s surplus is allowed to invest in a risk-free asset and a risky asset and the company’s risk is reduced through proportional reinsurance, while in addition the claim process is assumed to follow a Brownian motion with drift. By solving the corresponding Hamilton-Jacobi-Bellman equations, the optimal reinsurance-investment strategy is derived. The presented results generalize those by Taksar [1].     

Optimal dividend payout for classical risk model with risk constraint

In this paper we consider the problem of maximizing the total discounted utility of dividend payments for a Cramér-Lundberg risk model subject to both proportional and fixed transaction costs.We assume that dividend payments are prohibited unless the surplus of insurance company has reached a level b.Given fixed level b,we derive a integro-differential equation satisfied by the value function.By solving this equation we obtain the analytical solutions of the value function and the optimal dividend strategy when claims are exponentially distributed.Finally we show how the threshold b can be determined so that the expected ruin time is not less than some T.Also,numerical examples are presented to illustrate our results.     

含不动产项目的保险公司再保险-投资策略

站在保险公司管理者的角度, 考虑存在不动产项目投资机会时保险公司的再保险--投资策略问题. 假定保险公司可以投资于不动产项目、风险证券和无风险证券, 并通过比例再保险控制风险, 目标是最小化保险公司破产概率并求得相应最佳策略, 包括: 不动产项目投资时机、 再保险比例以及投资于风险证券的金额. 运用混合随机控制-最优停时方法, 得到最优值函数及最佳策略的显式解. 结果表明, 当且仅当其盈余资金多于某一水平(称为投资阈值)时保险公司投资于不动产项目. 进一步的数值算例分析表明: (a)~不动产项目投资的阈值主要受项目收益率影响而与投资金额无明显关系, 收益率越高则投资阈值越低; (b)~市场环境较好(牛市)时项目的投资阈值降低; 反之, 当市场环境较差(熊市)时投资阈值提高.     

On the analysis of the Gerber-Shiu discounted penalty function for risk processes with Markovian arrivals

In this paper, we consider an insurance risk model governed by a Markovian arrival claim process and by phase-type distributed claim amounts, which also allows for claim sizes to be correlated with the inter-claim times. A defective renewal equation of matrix form is derived for the Gerber-Shiu discounted penalty function and solved using matrix analytic methods. The use of the busy period distribution for the canonical fluid flow model is a key factor in our analysis, allowing us to obtain an explicit form of the Gerber-Shiu discounted penalty function avoiding thus the use of Lundberg’s fundamental equation roots. As a special case, we derive the triple Laplace transform of the time to ruin, surplus prior to ruin, and deficit at ruin in explicit form, further obtaining the discounted joint and marginal moments of the surplus prior to ruin and the deficit at ruin.     

On absolute ruin minimization under a diffusion approximation model

In this paper, we assume that the surplus process of an insurance entity is represented by a pure diffusion. The company can invest its surplus into a Black-Scholes risky asset and a risk free asset. We impose investment restrictions that only a limited amount is allowed in the risky asset and that no short-selling is allowed. We further assume that when the surplus level becomes negative, the company can borrow to continue financing. The ultimate objective is to seek an optimal investment strategy that minimizes the probability of absolute ruin, i.e. the probability that the liminf of the surplus process is negative infinity. The corresponding Hamilton-Jacobi-Bellman (HJB) equation is analyzed and a verification theorem is proved; applying the HJB method we obtain explicit expressions for the S-shaped minimal absolute ruin function and its associated optimal investment strategy. In the second part of the paper, we study the optimization problem with both investment and proportional reinsurance control. There the minimal absolute ruin function and the feedback optimal investment-reinsurance control are found explicitly as well.     

双险种最优再保险策略

本文对双险种风险模型,在一险种采取比例再保险,另一险种采取超出损失再保险策略下,得到调节系数与再保险自留水平之间的函数关系式,在理赔额为指数分布和Erlang(2)分布的条件下,得到最优比例再保险和超出损失再保险的自留水平,以及调节系数最大值。     

Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint

In this paper, the basic claim process is assumed to follow a Brownian motion with drift. In addition, the insurer is allowed to invest in a risk-free asset and n risky assets and to purchase proportional reinsurance. Under the constraint of no-shorting, we consider two optimization problems: the problem of maximizing the expected exponential utility of terminal wealth and the problem of minimizing the probability of ruin. By solving the corresponding Hamilton–Jacobi–Bellman equations, explicit expressions for their optimal value functions and the corresponding optimal strategies are obtained. In particular, when there is no risk-free interest rate, the results indicate that the optimal strategies, under maximizing the expected exponential utility and minimizing the probability of ruin, are equivalent for some special parameter. This validates Ferguson’s longstanding conjecture about the relation between the two problems.