Shuanming Li

On a general class of renewal risk process: analysis of the Gerber-Shiu function

We consider a compound renewal (Sparre Andersen) risk process with interclaim times that have a K n distribution (i.e. the Laplace transform of their density function is a ratio of two polynomials of degree at most n ∈ N). The Laplace transform of the expected discounted penalty function at ruin is derived. This leads to a generalization of the defective renewal equations given by Willmot (1999) and Gerber and Shiu (2005). Finally, explicit results are given for rationally distributed claim severities.

The Distribution of the Dividend Payments in theCompound Poisson Risk Model Perturbed byDiffusion

We consider a diffusion perturbed classical compound Poisson risk model in the presence of a constant dividend barrier. An integro-differential equation with certain boundary conditions for the n-th moment of the discounted dividend payments prior to ruin is derived and solved. Its solution can be expressed in terms of the expected discounted penalty (Gerber-Shiu) functions due to oscillation in the corresponding perturbed risk model without a barrier. When the discount factor δ is zero, we show that all the results can be expressed in terms of the non-ruin probability in the perturbed risk model without a barrier.

Moments of the Dividend Payments and Related Problems in a Markov-Modulated Risk Model

Abstract In this paper we derive some results on the dividend payments prior to ruin in a Markovmodulated risk process in which the rate for the Poisson claim arrival process and the distribution of the claim sizes vary in time depending on the state of an underlying (external) Markov jump process {J(t); t ≥ 0}. The main feature of the model is the flexibility in modeling the arrival process in the sense that periods with very frequent arrivals and periods with very few arrivals may alternate, and that the states of {J(t); t ≥ 0} could describe, for example, epidemic types in health insurance or weather conditions in car insurance. A system of integro-differential equations with boundary conditions satisfied by the nth moment of the present value of the total dividends prior to ruin, given the initial environment state, is derived and solved. We show that the probabilities that the surplus process attains a dividend barrier from the initial surplus without first falling below zero and the Laplace transforms of the time that the surplus process first hits a barrier without ruin occurring can be expressed in terms of the solution of the above-mentioned system of integro-differential equations. In the two-state model, explicit results are obtained when both claim amounts are exponentially distributed.

On a class of discrete time renewal risk models

We consider a class of compound renewal (Sparre Andersen) risk process with claim waiting times having a discrete K m distribution, i.e., the probability generating function (p.g.f.) of the distribution function is a ratio of two polynomials of order . The classical compound binomial risk model is a special case when m=1. A recursive formula is derived for the expected discounted penalty (Gerber-Shiu) function, which can be used to analyze many quantities associated with the time of ruin, e.g., the surplus before ruin, the deficit at ruin, and the claim causing ruin. Detailed discussions are given in two special cases: claim sizes are rationally distributed, or the claim sizes distribution has a finite support.We consider a class of compound renewal (Sparre Andersen) risk process with claim waiting times having a discrete K m distribution, i.e., the probability generating function (p.g.f.) of the distribution function is a ratio of two polynomials of order . The classical compound binomial risk model is a special case when m=1. A recursive formula is derived for the expected discounted penalty (Gerber-Shiu) function, which can be used to analyze many quantities associated with the time of ruin, e.g., the surplus before ruin, the deficit at ruin, and the claim causing ruin. Detailed discussions are given in two special cases: claim sizes are rationally distributed, or the claim sizes distribution has a finite support.

The Gerber–Shiu function in a Sparre Andersen risk process perturbed by diffusion

We consider a Sparre Andersen risk process that is perturbed by an independent diffusion process, in which claim inter-arrival times have a generalized Erlang(n) distribution (i.e. as the sum of n independent exponentials, with possibly different means). This leads to a generalization of the defective renewal equations for the expected discounted penalty function at the time of ruin given by Tsai and Willmot [10,11] and Gerber and Shiu [21,22]. The limiting behavior of the expected discounted penalty function is studied, when the dispersion coefficient goes to zero. Finally, explicit results are given for the case where n=2.

The Decompositions of the Discounted Penalty Functions and Dividends-Penalty Identity in a Markov-Modulated Risk Model

In this paper, we study the expected discounted penalty functions and their decompositions in a Markov-modulated risk process in which the rate for the Poisson claim arrivals and the distribution of the claim amounts vary in time depending on the state of an underlying (external) Markov jump process. The main feature of the model is the flexibility modeling the arrival process in the sense that periods with very frequent arrivals and periods with very few arrivals may alternate. Explicit formulas for the expected discounted penalty function at ruin, given the initial surplus, and the initial and terminal environment states, are obtained when the initial surplus is zero or when all the claim amount distributions are from the rational family. We also investigate the distributions of the maximum surplus before ruin and the maximum severity of ruin. The dividends-penalty identity is derived when the model is modified by applying a barrier dividend strategy.

The perturbed compound Poisson risk model with two-sided jumps

In this paper, we consider a perturbed compound Poisson risk model with two-sided jumps. The downward jumps represent the claims following an arbitrary distribution, while the upward jumps are also allowed to represent the random gains. Assuming that the density function of the upward jumps has a rational Laplace transform, the Laplace transforms and defective renewal equations for the discounted penalty functions are derived, and the asymptotic estimate for the probability of ruin is also studied for heavy-tailed downward jumps. Finally, some explicit expressions for the discounted penalty functions, as well as numerical examples, are given.

Distributions of the surplus before ruin, the deficit at ruin and the claim causing ruin in a class of discrete time risk models

We consider a class of compound renewal (Sparre Andersen) risk process with claim inter-arrival times having a discrete K m distribution, i.e., the probability generating function (p.g.f.) of its distribution function is a ratio of two polynomials of order . The classical compound binomial risk model is a special case when m=1. Li [16] gives a recursive formula for the Gerber-Shiu function. In this paper, an explicit formula for the Gerber-Shiu function is given in terms of a compound geometric distribution function, through which many ruin related quantities are analyzed, e.g., ruin probability, the p.g.f. of the time of ruin, joint and marginal distributions of the surplus before ruin, the deficit at ruin, and the claim causing ruin. Detailed discussions are given in two special cases: claim sizes are rationally distributed, or the claim sizes distribution has a finite support.We consider a class of compound renewal (Sparre Andersen) risk process with claim inter-arrival times having a discrete K m distribution, i.e., the probability generating function (p.g.f.) of its distribution function is a ratio of two polynomials of order . The classical compound binomial risk model is a special case when m=1. Li [16] gives a recursive formula for the Gerber-Shiu function. In this paper, an explicit formula for the Gerber-Shiu function is given in terms of a compound geometric distribution function, through which many ruin related quantities are analyzed, e.g., ruin probability, the p.g.f. of the time of ruin, joint and marginal distributions of the surplus before ruin, the deficit at ruin, and the claim causing ruin. Detailed discussions are given in two special cases: claim sizes are rationally distributed, or the claim sizes distribution has a finite support.

A review of discrete-time risk models

In this paper, we present a review of results for discrete-time risk models, including the compound binomial risk model and some of its extensions. While most theoretical risk models use the concept of time continuity, the practical reality is discrete. For instance, recursive formulas for discretetime models can be obtained without assuming a claim severity distribution and are readily programmable in practice. Hence themodels, techniques used, and results reviewed here for discrete-time risk models are of independent scientific interest. Yet, results for discrete-time risk models can give, in addition, a simpler understanding of their continuous-time analogue. For example, these results can serve as approximations or bounds for the corresponding results in continuous-time models. This paper will serve as a detailed reference for the study of discrete-time risk models.ResumenEn este artículo hacemos un repaso de los resultados para modelos de riesgo en tiempo discreto, incluyendo el modelo de riesgo binomial-compuesto, así como algunas de sus extensiones. Aunque gran parte de los modelos teóricos de riesgo se basen en el concepto de continuidad del tiempo, la realidad práctica es en sí discreta. Por ejemplo, en la práctica actuarial se programan fórmulas recursivas para modelos en tiempo discreto, sin necesidad de suponer una distribución de pérdidas conocida. Con lo cual estos modelos, las técnicas y los resultados que listamos para modelos de riesgo en tiempo discreto, generan un cierto interés científico propio. Pero más allá de sus aplicaciones directas, estos resultados para modelos en tiempo discreto también proporcionan un camino más simple hacia los modelos de riesgo análogos en tiempo continuo. Por ejemplo, los resultados en tiempo discreto pueden servir de aproximaciones o de cotas para sus resultados correspondientes en tiempo continuo. El propósito de este artículo es que pueda servir de referencia detallada para el estudio de modelos de riesgo en tiempo discreto.

Ruin Probabilities for Two Classes of Risk Processes

We consider a risk model with two independent classes of insurance risks. We assume that the two independent claim counting processes are, respectively, Poisson and Sparre Andersen processes with generalized Erlang(2) claim interarrival times. The Laplace transform of the non-ruin probability is derived from a system of integro-differential equations. Explicit results can be obtained when the initial reserve is zero and the claim severity distributions of both classes belong to the K n family of distributions. A relation between the ruin probability and the distribution of the supremum before ruin is identified. Finally, the Laplace transform of the non-ruin probability of a perturbed Sparre Andersen risk model with generalized Erlang(2) claim inter-arrival times is derived when the compound Poisson process converges weakly to a Wiener process.