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.     

Two-sided bounds for the distribution of the deficit at ruin in the renewal risk model

We obtain lower and upper bounds for the severity of ruin in the renewal (Sparre Andersen) model of risk theory. We present two types of bounds: (i) bounds applicable generally; and (ii) exponential bounds for the case where the adjustment coefficient of the risk process exists. Many of these bounds are obtained using existing bounds and the integral equation for the severity of ruin.     

Asymptotic ruin probabilities for a multidimensional renewal risk model with multivariate regularly varying claims

This paper studies a continuous-time multidimensional risk model with constant force of interest and dependence structures among random factors involved. The model allows a general dependence among the claim-number processes from different insurance businesses. Moreover, we utilize the framework of multivariate regular variation to describe the dependence and heavy-tailed nature of the claim sizes. Some precise asymptotic expansions are derived for both finite-time and infinite-time ruin probabilities.     

Tail behavior of random sums under consistent variation with applications to the compound renewal risk model

In this paper, we consider the random sums of i.i.d. random variables ξ ₁,ξ ₂,... with consistent variation. Asymptotic behavior of the tail P(ξ₁ + ... + ξ_η > x), where η is independent of ξ ₁,ξ ₂,..., is obtained for different cases of the interrelationships between the tails of ξ ₁ and η. Applications to the asymptotic behavior of the finite-time ruin probability ψ(x,t) in a compound renewal risk model, earlier introduced by Tang et al. (Stat Probab Lett 52, 91–100 (2001)), are given. The asymptotic relations, as initial capital x increases, hold uniformly for t in a corresponding region. These asymptotic results are illustrated in several examples.      

Collective risk models with dependence

In actuarial science, collective risk models, in which the aggregate claim amount of a portfolio is defined in terms of random sums, play a crucial role. In these models, it is common to assume that the number of claims and their amounts are independent, even if this might not always be the case. We consider collective risk models with different dependence structures. Due to the importance of such risk models in an actuarial setting, we first investigate a collective risk model with dependence involving the family of multivariate mixed Erlang distributions. Other models based on mixtures involving bivariate and multivariate copulas in a more general setting are then presented. These different structures allow to link the number of claims to each claim amount, and to quantify the aggregate claim loss. Then, we use Archimedean and hierarchical Archimedean copulas in collective risk models, to model the dependence between the claim number random variable and the claim amount random variables involved in the random sum. Such dependence structures allow us to derive a computational methodology for the assessment of the aggregate claim amount. While being very flexible, this methodology is easy to implement, and can easily fit more complicated hierarchical structures.     

Ruin measures for a compound Poisson risk model with dependence based on the Spearman copula and the exponential claim sizes

This paper is devoted to an extension to the classical compound risk model. We relax the independence assumption of claim amounts and interclaim times. The dependent structure between these random variables is described by the Spearman copula. We study the Laplace transform of the discounted penalty function and we give the explicit expression of it for the exponential claim size.     

A risk model with renewal shot-noise Cox process

In this paper we generalise the risk models beyond the ordinary framework of affine processes or Markov processes and study a risk process where the claim arrivals are driven by a Cox process with renewal shot-noise intensity. The upper bounds of the finite-horizon and infinite-horizon ruin probabilities are investigated and an efficient and exact Monte Carlo simulation algorithm for this new process is developed. A more efficient estimation method for the infinite-horizon ruin probability based on importance sampling via a suitable change of probability measure is also provided; illustrative numerical examples are also provided.     

The periodic risk model with investment

We consider a periodic risk model with the possibility of investing into a risky asset, given by a geometrical Brownian motion. The aim is to maximize the adjustment coefficient of the risk process. It is shown that the optimal investment strategy only depends on the averaged data of the model and is constant over time. Thus maximizing the adjustment coefficient is a very weak optimization criterion.     

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.     

Ruin theory in the linear model

The probability of ruin is examined in a model where the annual gains of an insurance company are dependent random variables. The model used is the linear model (well known in time-series analysis) which includes the autoregressive model and the moving average model as special cases. It is also shown that a certain credibility model can be interpreted as a first-order model of the mixed type.     

On the renewal risk model under a threshold strategy

In this paper, we consider the renewal risk process under a threshold dividend payment strategy. For this model, the expected discounted dividend payments and the Gerber–Shiu expected discounted penalty function are investigated. Integral equations, integro-differential equations and some closed form expressions for them are derived. When the claims are exponentially distributed, it is verified that the expected penalty of the deficit at ruin is proportional to the ruin probability.     

Tail bounds for the distribution of the deficit in the renewal risk model

We obtain upper and lower bounds for the tail of the deficit at ruin in the renewal risk model, which are (i) applicable generally; and (ii) based on reliability classifications. We also derive two-side bounds, in the general case where a function satisfies a defective renewal equation, and we apply them to the renewal model, using the function Λ_u introduced by [Psarrakos, G., Politis, K., 2007. A generalisation of the Lundberg condition in the Sparre Andersen model and some applications (submitted for publication)]. Finally, we construct an upper bound for the integrated function and an asymptotic result when the adjustment coefficient exists.     

A time-series risk model with constant interest for dependent classes of business

In this paper, we propose a discrete-time model with dependent classes of business using a time-series approach. Specifically, premiums and claims of all classes are supposed to satisfy a multivariate first-order autoregressive time-series model. A constant interest rate is also included in the model. A Lundberg-type inequality for the ruin probability is deduced. We also give an example with constant premiums and two classes of claims for which an expression as well as an exponential bound for the ruin probability is given. A simulation study is provided to help understanding the model.     

The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims

Recently, Tang [Tang, Q., 2005a. Asymptotic ruin probabilities of the renewal model with constant interest force and regular variation. Scand. Actuar. J. (1), 1–5] obtained a simple asymptotic formula for the ruin probability of the renewal risk model with constant interest force and regularly varying tailed claims. In this paper, we use a completely different approach to extend Tang’s result to the case in which the claims are pairwise negatively dependent and extended regularly varying tailed.     

On a dual model with a dividend threshold

In insurance mathematics, a compound Poisson model is often used to describe the aggregate claims of the surplus process. In this paper, we consider the dual of the compound Poisson model under a threshold dividend strategy. We derive a set of two integro-differential equations satisfied by the expected total discounted dividends until ruin and show how the equations can be solved by using only one of the two integro-differential equations. The cases where profits follow an exponential or a mixture of exponential distributions are then solved and the discussion for the case of a general profit distribution follows by the use of Laplace transforms. We illustrate how the optimal threshold level that maximizes the expected total discounted dividends until ruin can be obtained, and finally we generalize the results to the case where the surplus process is a more general skip-free downwards Lévy process.     

Uniform asymptotics for the ruin probabilities in a bidimensional renewal risk model with strongly subexponential claims

This paper considers a bidimensional continuous-time renewal risk model of insurance business with different claim-number processes and strongly subexponential claims. For the finite-time ruin probability defined as the probability for the aggregate surplus process to break down the horizontal line at the level zero within a given time, an uniform asymptotic formula is established, which provides new insights into the solvency ability of the insurance company.     

On the total operating costs up to default in a renewal risk model

The paper proposes a new approach to study a general class of ruin-related quantities in the context of a renewal risk model. While the classical approaches in Sparre Andersen models have their own merits, the approach presented in this paper has its advantages from the following perspectives. (1) The underlying surplus process has the flexibility to reflect a broad range of scenarios for surplus growth including dividend policies and interest returns. (2) The solution method provides a general framework to unify a great variety of existing ruin-related quantities such as Gerber–Shiu functions and the expected present value of dividends paid up to ruin, and facilitates derivations of new ruin-related quantities such as the expected present value of total claim costs up to ruin, etc. In the end, many specific examples are explored to demonstrate its application in renewal risk models.     

Asymptotic finite-time ruin probability for a bidimensional renewal risk model with constant interest force and dependent subexponential claims

This paper considers a bidimensional renewal risk model with constant interest force and dependent subexponential claims. Under the assumption that the claim size vectors form a sequence of independent and identically distributed random vectors following a common bivariate Farlie–Gumbel–Morgenstern distribution, we derive for the finite-time ruin probability an explicit asymptotic formula.     

On a compound Poisson risk model with dependence and in the presence of a constant dividend barrier

In this paper, we consider a classical risk process with dependence and in the presence of a constant dividend barrier. The dependence structure between the claim amounts and the interclaim times is introduced through a Farlie–Gumbel–Morgenstern copula. We analyze the expectation of the discounted penalty function and the expectation of the present value of the distributed dividends. For each function, an integro‐differential equation with boundary conditions is derived, and the solution is provided. Finally, we find an explicit solution for each function when the claim amounts are exponentially distributed. We illustrate the impact of the dependence on these two quantities. Copyright © 2012 John Wiley & Sons, Ltd.     

The net Bayes premium with dependence between the risk profiles

In Bayesian analysis it is usual to assume that the risk profiles Θ₁ and Θ₂ associated with the random variables “number of claims” and “amount of a single claim”, respectively, are independent. A few studies have addressed a model of this nature assuming some degree of dependence between the two random variables (and most of these studies include copulas). In this paper, we focus on the collective and Bayes net premiums for the aggregate amount of claims under a compound model assuming some degree of dependence between the random variables Θ₁ and Θ₂. The degree of dependence is modelled using the Sarmanov–Lee family of distributions [Sarmanov, O.V., 1966. Generalized normal correlation and two-dimensional Frechet classes. Doklady (Soviet Mathematics) 168, 596–599 and Ting-Lee, M.L., 1996. Properties and applications of the Sarmanov family of bivariate distributions. Communications Statistics: Theory and Methods 25 (6) 1207–1222], which allows us to study the impact of this assumption on the collective and Bayes net premiums. The results obtained show that a low degree of correlation produces Bayes premiums that are highly sensitive.