Gordon E. Willmot

Analysis of a defective renewal equation arising in ruin theory

Abstract This paper studies in detail the solution of a defective renewal equation which involves the time of ruin, the surplus immediately before ruin, and the deficit at the time of ruin. The analysis is simplified by introduction and analysis of a related compound geometric distribution, which is studied in detail. Tijms approximations and bounds for these quantities are also discussed. Examples are given for the cases when the claim size distribution is exponential, combinations of exponentials and mixtures of Erlangs. In a subsequent paper, we will extend our analysis to the moments of the time of ruin, the moments of the surplus before the time of ruin, the moments of the deficit at the time of ruin, and correlations between them.

The moments of the time of ruin, the surplus before ruin, and the deficit at ruin

Abstract In this paper we extend the results in Lin and Willmot (1999 Insurance: Mathematics and Economics 25, 63–84) to properties related to the joint and marginal moments of the time of ruin, the surplus before the time of ruin, and the deficit at the time of ruin. We use an approach developed in Lin and Willmot (1999) , under which the solution to a defective renewal equation is expressed in terms of a compound geometric tail, to derive explicitly the joint and marginal moments. This approach also allows for the establishment of recursive relations between these moments. Examples are given for the cases when the claim size distribution is exponential, combinations of exponentials and mixtures of Erlangs.

The Poisson-Inverse Gaussian distribution as an alternative to the negative binomial

Abstract The basic distributional properties and estimation techniques of the Poisson-Inverse Gaussian (P-IG) distribution are reviewed. Its use both as a mixed and compound claim frequency model are also discussed, as well as a review of the aggregate claims distribution when the P-IG is the claim frequency component. The many properties which are analogous to those of the negative binomial are emphasized, and the superior fit to automobile claim frequency data is demonstrated. The P-IG merits consideration as a model for claim frequency data due to its good fit to data, physical justification, and its abundance of convenient mathematical properties.

A generalized defective renewal equation for the surplus process perturbed by diffusion

Abstract In this paper, we consider the surplus process of the classical continuous time risk model containing an independent diffusion (Wiener) process. We generalize the defective renewal equation for the expected discounted function of a penalty at the time of ruin in Garber and Landry [Insurance: Math. Econ. 22 (1998) 263]. Then an asymptotic formula for the expected discounted penalty function is proposed. In addition, the associated claim size distribution is studied, and reliability-based class implications for the distribution are given.

Ruin probabilities in the compound binomial model

Abstract In this paper explicit formulas are derived for finite time ruin probabilities in the discrete time and state-space compound binomial model using the technique of generating functions. Ultimate ruin probabilities are then obtained, and a close connection is extablished with the ultimate ruin probabilities in the usual compound Poisson model when the claim severity distribution is a (truncated) mixed Poisson distribution.

On the Class of Erlang Mixtures with Risk Theoretic Applications

Abstract A wide variety of distributions are shown to be of mixed-Erlang type. Useful computational formulas result for many quantities of interest in a risk-theoretic context when the claim size distribution is an Erlang mixture. In particular, the aggregate claims distribution and related quantities such as stop-loss moments are discussed, as well as ruin-theoretic quantities including infinitetime ruin probabilities and the distribution of the deficit at ruin. A very useful application of the results is the computation of finite-time ruin probabilities, with numerical examples given. Finally, extensions of the results to more general gamma mixtures are briefly examined.

On recursive evaluation of mixed poisson probabilities and related quantities

Abstract Recursive formulae are derived for the probabilities of a wide variety of mixed Poisson distributions. Known results are unified and extended. Related formulae are discussed for transformed mixing random variables, shifted and truncated mixing distributions, compound distributions, and tail probabilities. Applications of these models are briefly discussed.

The total claims distribution under inflationary conditions

Abstract The total claims distribution over a fixed period of time with time dependent claim amounts is considered. A representation for the associated density function is found under certain conditions, including the important case with Poisson or mixed Poisson claim number processes and constant inflation. Methods of evaluation of this density are considered, and the cases with exponential claim sizes and regular variation of the tail are discussed in more detail.

ON THE EXPECTATION OF TOTAL DISCOUNTED OPERATING COSTS UP TO DEFAULT AND ITS APPLICATIONS

In this paper we first consider the expectation of the total discounted claim costs up to the time of ruin, and then, more generally, we study the expectation of the total discounted operating costs up to the time of default, which is the first passage time of a surplus process downcrossing a given level. These two quantities include the expected discounted penalty function at ruin or the Gerber–Shiu function, the expected total discounted dividends up to ruin, and other interesting quantities as special cases among a class of risk processes. As an illustration, we consider a piecewise-deterministic compound Poisson risk model. This model recovers many risk models appearing in the literature such as the compound Poisson risk models with interest, absolute ruin, dividends, multiple thresholds, and their dual models. We derive and solve the integro-differential equation for the expected present value of the total discounted operating costs up to default. The solutions to the expected present value of the total discounted operating costs up to default can be used as a unified approach to solving many ruin-related quantities. As applications, we derive explicit solutions for the expected accumulated utility up to ruin, the absolute ruin probability with varying borrowing rates, the expected total discounted claim costs up to ruin, the Gerber–Shiu function with two-sided jumps, and the price for a perpetual American put option with two-sided jumps.

The Gerber-Shiu discounted penalty function in the stationary renewal risk model

Abstract The discounted penalty function introduced by Gerber and Shiu [North American Actuarial Journal 2 (1998) 48] is considered in the stationary renewal risk model, where it is expressed in terms of the same discounted penalty function in the ordinary renewal risk model. This relationship unifies and generalizes known special cases. An invariance property between the stationary renewal risk model and the classical Poisson model with respect to the ruin probability is also generalized as a result.