Insurance Risk Model Research Articles

Occupation times of intervals until first passage times for spectrally negative Lévy processes

In this paper, we identify Laplace transforms of occupation times of intervals until first passage times for spectrally negative Lévy processes. New analytical identities for scale functions are derived and therefore the results are explicitly stated in terms of the scale functions of the process. Applications to option pricing and insurance risk models are also presented.

A note on the max-sum equivalence of randomly weighted sums of heavy-tailed random variables

This paper investigates the asymptotic behavior for the tail probability of the randomly weighted sums ∑k=1nθkXk and their maximum, where the random variables Xk and the random weights θk follow a certain dependence structure proposed by Asimit and Badescu [1] and Li et al. [2]. The obtained results can be used to obtain asymptotic formulas for ruin probability in the insurance risk models with discounted factors.

Precise large deviations for actual aggregate loss process in a dependent compound customer-arrival-based insurance risk model

In this paper, we investigate a dependent compound customer-arrival-based insurance risk model, in which the kth customer purchases a random number of insurance contracts, his/her actual individual claim sizes are described as negatively dependent consistently varying-tailed random variables multiplied by a general shot noise function, and the individual customer-arrival process is a Poisson process. We obtain some precise large deviation results for the actual aggregate loss process, which extend and close the gaps of the related results of Shen et al. [X. Shen, Z. Lin, and Y. Zhang, Precise large deviations for the actual aggregate loss process, Stoch. Anal. Appl., 27:1000–1013, 2009].

A unified analysis of claim costs up to ruin in a Markovian arrival risk model

An insurance risk model where claims follow a Markovian arrival process (MArP) is considered in this paper. It is shown that the expected present value of total operating costs up to default H, as a generalization of the classical Gerber–Shiu function, contains more non-trivial quantities than those covered in Cai et al. (2009), such as all moments of the discounted claim costs until ruin. However, it does not appear that the Gerber–Shiu function ϕ with a generalized penalty function which additionally depends on the surplus level immediately after the second last claim before ruin (Cheung et al., 2010a) is contained in H. This motivates us to investigate an even more general function Z from which both H and ϕ can be retrieved as special cases. Using a matrix version of Dickson–Hipp operator (Feng, 2009b), it is shown that Z satisfies a Markov renewal equation and hence admits a general solution. Applications to other related problems such as the matrix scale function, the minimum and maximum surplus levels before ruin are given as well.

Some results on absolute ruin in the perturbed insurance risk model with investment and debit interests

In this paper, we consider a perturbed compound Poisson risk model with investment and debit interests. Dividends are paid to the shareholders according to a threshold dividend strategy. An alternative assumption is that when the surplus is negative, a debit interest is applied and when the surplus is above a certain positive level, the insurer could earn investment interest. Integro-differential equations with boundary conditions satisfied by the moment-generating function, the nth moment of the present value of all dividends until absolute ruin and the Gerber–Shiu expected discounted penalty function are obtained. Then, we present the explicit expressions for the zero discounted nth moment of the present value of all dividends until absolute ruin in the case of exponential claims. Finally, numerical example is also given to illustrate the effect of the related parameters on the first moment of the present value of all dividends until absolute ruin.

An Insurance Risk Model with Parisian Implementation Delays

We consider a similar variant of the event ruin for a Lévy insurance risk process as in Czarna and Palmowski (J Appl Probab 48(4):984–1002, 2011) and Loeffen et al. (to appear, 2011) when the surplus process is allowed to spend time under a pre-specified default level before ruin is recognized. In these two articles, the ruin probability is examined when deterministic implementation delays are allowed. In this paper, we propose to capitalize on the idea of randomization and thus assume these delays are of a mixed Erlang nature. Together with the analytical interest of this problem, we will show through the development of new methodological tools that these stochastic delays lead to more explicit and computable results for various ruin-related quantities than their deterministic counterparts. Using the modern language of scale functions, we study the Laplace transform of this so-called Parisian time to ruin in an insurance risk model driven by a spectrally negative Lévy process of bounded variation. In the process, a generalization of the two-sided exit problem for this class of processes is further obtained.

The Finite Time Ruin Probability of a New Risk Model Based on Entrance Process

In this article, we consider a new insurance risk model based on the entrance process proposed in Li et al. (2005), and investigate the finite time ruin probabilities of this model. It is showed that an exponential upper bound for the finite time ruin probability exists, when the distributions of the claim size are light tailed. Furthermore, when the distributions of the claim size are heavy tailed, an asymptotic formula for the finite time ruin probability is obtained.

Randomized Dividends in a Discrete Insurance Risk Model with Stochastic Premium Income

The compound binomial insurance risk model is extended to the case where the premium income process, based on a binomial process, is no longer a constant premium rate of 1 per period and insurer pays a dividend of 1 with a probabilityq0when the surplus is greater than or equal to a nonnegative integerb. The recursion formulas for expected discounted penalty function are derived. As applications, we present the recursion formulas for the ruin probability, the probability function of the surplus prior to the ruin time, and the severity of ruin. Finally, numerical example is also given to illustrate the effect of the related parameters on the ruin probability.

Exact and Asymptotic Results for Insurance Risk Models with Surplus-dependent Premiums

In this paper we develop a symbolic technique to obtain asymptotic expressions for ruin probabilities and discounted penalty functions in renewal insurance risk models when the premium income depends on the present surplus of the insurance portfolio. The analysis is based on boundary problems for linear ordinary differential equations with variable coefficients. The algebraic structure of the Green's operators allows us to develop an intuitive way of tackling the asymptotic behavior of the solutions, leading to exponential-type expansions and Cramér-type asymptotics. Furthermore, we obtain closed-form solutions for more specific cases of premium functions in the compound Poisson risk model.

On the Expected Discounted Penalty Function for a Markov Regime-Switching Insurance Risk Model with Stochastic Premium Income

We consider a Markovian regime-switching risk model (also called the Markov-modulated risk model) with stochastic premium income, in which the premium income and the claim occurrence are driven by the Markovian regime-switching process. The purpose of this paper is to study the integral equations satisfied by the expected discounted penalty function. In particular, the discount interest force process is also regulated by the Markovian regime-switching process. Applications of the integral equations are given to be the Laplace transform of the time of ruin, the deficit at ruin, and the surplus immediately before ruin occurs. For exponential distribution, the explicit expressions for these quantities are obtained. Finally, a numerical example is also given to illustrate the effect of the related parameters on these quantities.

Studies on a Double Poisson-Geometric Insurance Risk Model with Interference

This paper mainly studies a generalized double Poisson-Geometric insurance risk model. By martingale and stopping time approach, we obtain adjustment coefficient equation, the Lundberg inequality, and the formula for the ruin probability. Also the Laplace transformation of the time when the surplus reaches a given level for the first time is discussed, and the expectation and its variance are obtained. Finally, we give the numerical examples.

A Note on the Tail Behavior of Randomly Weighted Sums with Convolution-Equivalently Distributed Random Variables

We investigate the tailed asymptotic behavior of the randomly weighted sums with increments with convolution-equivalent distributions. Our obtained result can be directly applied to a discrete-time insurance risk model with insurance and financial risks and derive the asymptotics for the finite-time probability of the above risk model.

Large deviations for random sums of differences between two sequences of random variables with applications to risk theory

This paper investigates some precise large deviations for the random sums of the differences between two sequences of independent and identically distributed random variables, where the minuend random variables have subexponential tails, and the subtrahend random variables have finite second moments. As applications to risk theory, the customer-arrival-based insurance risk model is considered, and some uniform asymptotics for the ruin probabilities of an insurance company are derived as the number of customers or the time tends to infinity. MSC: 60F10; 62E20; 62P05

Extreme value theorems of uncertain process with application to insurance risk model

Uncertain process is a sequence of uncertain variables indexed by time. This paper presents a series of extreme value theorem of uncertain independent increment process and provides uncertainty distribution of first hitting time. This paper also proposes an insurance risk model with uncertain claims. Finally, a concept of ruin index is defined and a ruin index formula is given.

Asymptotic results for renewal risk models with risky investments

We consider a renewal jump–diffusion process, more specifically a renewal insurance risk model with investments in a stock whose price is modeled by a geometric Brownian motion. Using Laplace transforms and regular variation theory, we introduce a transparent and unifying analytic method for investigating the asymptotic behavior of ruin probabilities and related quantities, in models with light- or heavy-tailed jumps, whenever the distribution of the time between jumps has rational Laplace transform.

Tail behavior of the product of two dependent random variables with applications to risk theory

Let the random vector (X,Y) follow a bivariate Sarmanov distribution, where X is real-valued and Y is nonnegative. In this paper we investigate the impact of such a dependence structure between X and Y on the tail behavior of their product Z = XY. When X has a regularly varying tail, we establish an asymptotic formula, which extends Breiman’s theorem. Based on the obtained result, we consider a discrete-time insurance risk model with dependent insurance and financial risks, and derive the asymptotic and uniformly asymptotic behavior for the (in)finite-time ruin probabilities.

Precise large deviations for compound random sums in the presence of dependence structures

In this paper, we deal with the compound random sums of dependent real-valued random variables with heavy-tailed distributions. We establish a precise large deviation result for a nonstandard renewal risk model in which innovations are extended negatively dependent real-valued random variables with a common dominatedly varying distribution function, their interarrival times are extended negatively dependent nonnegative random variables, and the numbers of innovations caused by individual events are also extended negatively dependent positive random variables. As an illustration of the obtained result, we give two applications related to some insurance risk models.

On a Generalization from Ruin to Default in a Lévy Insurance Risk Model

In a variety of insurance risk models, ruin-related quantities in the class of expected discounted penalty function (EDPF) were known to satisfy defective renewal equations that lead to explicit solutions. Recent development in the ruin literature has shown that similar defective renewal equations exist for a more general class of quantities than that of EDPF. This paper further extends the analysis of this new class of functions in the context of a spectrally negative Levy risk model. In particular, we present an operator-based approach as an alternative analytical tool in comparison with fluctuation theoretic methods used for similar quantities in the current literature. The paper also identifies a sufficient and necessary condition under which the classical results from defective renewal equation and those from fluctuation theory are interchangeable. As a by-product, we present a series representation of scale function as well as potential measure in terms of compound geometric distribution.

The Finite-Time Ruin Probability with Dependent Insurance and Financial Risks

Consider a discrete-time insurance risk model. Within period i, the net insurance loss is denoted by a real-valued random variable Xi. The insurer makes both risk-free and risky investments, leading to an overall stochastic discount factor Yi from time i to time i − 1. Assume that (Xi, Yi), i ∈ N, form a sequence of independent and identically distributed random pairs following a common bivariate Farlie-Gumbel-Morgenstern distribution with marginal distribution functions F and G. When F is subexponential and G fulfills some constraints in order for the product convolution of F and G to be subexponential too, we derive a general asymptotic formula for the finite-time ruin probability. Then, for special cases in which F belongs to the Fréchet or Weibull maximum domain of attraction, we improve this general formula to be transparent.

The Finite-Time Ruin Probability with Dependent Insurance and Financial Risks

Consider a discrete-time insurance risk model. Within periodi, the net insurance loss is denoted by a real-valued random variableXi. The insurer makes both risk-free and risky investments, leading to an overall stochastic discount factorYifrom timeito timei− 1. Assume that (Xi,Yi),i∈N, form a sequence of independent and identically distributed random pairs following a common bivariate Farlie-Gumbel-Morgenstern distribution with marginal distribution functionsFandG. WhenFis subexponential andGfulfills some constraints in order for the product convolution ofFandGto be subexponential too, we derive a general asymptotic formula for the finite-time ruin probability. Then, for special cases in whichFbelongs to the Fréchet or Weibull maximum domain of attraction, we improve this general formula to be transparent.