Andrei L. Badescu

Extremes on the Discounted Aggregate Claims in a Time Dependent Risk Model

This paper presents an extension of the classical compound Poisson risk model for which the inter-claim time and the forthcoming claim amount are no longer independent random variables (rvs). Asymptotic tail probabilities for the discounted aggregate claims are presented when the force of interest is constant and the claim amounts are heavy tail distributed rvs. Furthermore, we derive asymptotic finite time ruin probabilities, as well as asymptotic approximations for some common risk measures associated with the discounted aggregate claims. A simulation study is performed in order to validate the results obtained in the free interest risk model.

Risk processes analyzed as fluid queues

This paper presents the Laplace transform of the time until ruin for a fairly general risk model. The model includes both the classical and most Sparre-Andersen risk models with phase-distributed claim amounts as special cases. It also allows for correlated arrival processes, and claim sizes that depend upon environmental factors such as periods of contagion. The paper exploits the relationship between the surplus process and fluid queues, where a number of recent developments have provided the basis for our analysis.

Phase-type approximations to finite-time ruin probabilities in the Sparre-Andersen and stationary renewal risk models

The present paper extends the “Erlangization” idea introduced by Asmussen, Avram, and Usabel (2002) to the Sparre-Andersen and stationary renewal risk models. Erlangization yields an asymptotically-exact method for calculating finite time ruin probabilities with phase-type claim amounts. The method is based on finding the probability of ruin prior to a phase-type random horizon, independent of the risk process. When the horizon follows an Erlang-l distribution, the method provides a sequence of approximations that converges to the true finite-time ruin probability as l increases. Furthermore, the random horizon is easier to work with, so that very accurate probabilities of ruin are obtained with comparatively little computational effort. An additional section determines the phase-type form of the deficit at ruin in both models. Our work exploits the relationship to fluid queues to provide effective computational algorithms for the determination of these quantities, as demonstrated by the numerical examples.

On the analysis of a multi-threshold Markovian risk model

We consider a class of Markovian risk models perturbed by a multiple threshold dividend strategy in which the insurer collects premiums at rate c i whenever the surplus level resides in the i-th surplus layer, i=1, 2, …,n+1 where n<∞. We derive the Laplace-Stieltjes transform (LST) of the distribution of the time to ruin as well as the discounted joint density of the surplus prior to ruin and the deficit at ruin. By interpreting that the insurer, whose gross premium rate is c, pays dividends continuously at rate d i =c−c i whenever the surplus level resides in the i-th surplus layer, we also derive the expected discounted value of total dividend payments made prior to ruin. Our results are obtained via a recursive approach which makes use of an existing connection, linking an insurers surplus process to an embedded fluid flow process.

Analysis of a threshold dividend strategy for a MAP risk model

We consider a class of Markovian risk models in which the insurer collects premiums at rate c1(c2) whenever the surplus level is below (above) a constant threshold level b. We derive the Laplace-Stieltjes transform (LST) of the distribution of the time to ruin as well as the LST (with respect to time) of the joint distribution of the time to ruin, the surplus prior to ruin, and the deficit at ruin. By interpreting that the insurer pays dividends continuously at rate c1−c2 whenever the surplus level is above b, we also derive the expected discounted value of total dividend payments made prior to ruin. Our results are obtained by making use of an existing connection which links an insurers surplus process to an embedded fluid flow process.

The surplus prior to ruin and the deficit at ruin for a correlated risk process

This paper presents an explicit characterization for the joint probability density function of the surplus immediately prior to ruin and the deficit at ruin for a general risk process, which includes the Sparre-Andersen risk model with phase-type inter-claim times and claim sizes. The model can also accommodate a Markovian arrival process which enables claim sizes to be correlated with the inter-claim times. The marginal density function of the surplus immediately prior to ruin is specifically considered. Several numerical examples are presented to illustrate the application of this result.

Recursive Calculation of the Dividend Moments in a Multi-threshold Risk Model

Abstract In this article, we consider the class of risk models with Markovian claim arrivals studied by Badescu et al. (2005) and Ramaswami (2006), among others. Under a multi-threshold dividend structure, we develop a recursive algorithm for the calculation of the moments of the discounted dividend payments before ruin. Capitalizing on the connection between an insurer’s surplus process and its corresponding fluid flow process, our approach generalizes results obtained by Albrecher and Hartinger (2007) and Zhou (2006) in the framework of the classical compound Poisson risk model (with phase-type claim sizes). Contrary to the traditional analysis of the discounted dividend payments in risk theory, we develop a sample-path-analysis procedure that allows the determination of these moments with or without ruin occurrence (separately). Numerical examples are then considered to illustrate our main results and show the contribution of each component to the moments of the discounted dividend payments.

Modeling Correlated Frequencies with Application in Operational Risk Management

In this paper, we propose a copula-free approach for modeling correlated frequency distributions using an Erlang-based multivariate mixed Poisson distribution. We investigate some of the properties possessed by this class of distributions and derive a tailormade expectation-maximization algorithm for fitting purposes. The applicability of the proposed distribution is illustrated in an operational risk management context, where this class is used to model the operational loss frequencies and their complex dependence structure in a high-dimensional setting. Furthermore, by assuming that operational loss severities follow the mixture of Erlang distributions, our approach leads to a closed-form expression for the total aggregate loss distribution and its value-at-risk can be calculated easily by any numerical method. The efficiency and accuracy of the proposed approach are analyzed using a modified real operational loss data set.

On a Generalization of the Risk Model with Markovian Claim Arrivals

The class of risk models with Markovian arrival process (MAP) (see e.g., Neuts[ 15 ]) is generalized by allowing the waiting times between two successive events (which can be a change in the environmental state and/or a claim arrival) to have an arbitrary distribution. Using a probabilistic approach, we determine the solution for a class of Gerber–Shiu functions apart from some unknown constants when claim sizes have a mixed exponential distribution. Such constants are later determined using the more classic ruin-analytic approach. A numerical example is later considered to illustrate the tractability of the suggested methodology in the study of Gerber–Shiu functions.

“The Discounted Joint Distribution of the Surplus Prior to Ruin and the Deficit at Ruin in a Sparre Andersen Model,” Jiandong Ren, July 2007

CHEUNG, ERIC C. K. 2007. Discussion of ‘‘Moments of the Dividend Payments and Related Problems in a Markov-Modulated Risk Model.’’ North American Actuarial Journal 11(4): 145–48. GERBER, HANS U., AND ELIAS S. W. SHIU. 1998. On the Time Value of Ruin. North American Actuarial Journal 2(1): 48–72. KO, BANGWON. 2007. Discussion of ‘‘The Discounted Joint Distribution of the Surplus Prior to Ruin and the Deficit at Ruin in a Sparre Andersen Model.’’ North American Actuarial Journal 11(3): 136–37.