(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

The book reviews section generally accepts for review only those books whose content and level reflect the general editorial policy of Technometrics. Publishers are invited to send books for review to Eric R. Ziegel, Amoco Research Center, Mail Station F-l/C& P.O. Box 3011, Naperville, Illinois 60566-7011. Please include the price of the book. The opinions expressed in this section are those of the reviewers and do not necessarily reflect those of the editorial staff or the sponsoring societies. Listed prices were provided by the publisher when the books were received by Technometrics and may not be current.

Adventures in Stochastic Processes

Stochastic Models, an Algorithmic Approach , by Henk C. Tijms (Chichester: Wiley, 1994), 375 pages, paperback.

In today’s world of financial uncertainty, one major public concern is to assess (and possibly improve) the stability of companies that take on risks. Actuaries have been aware of that issue for a very long time and have a great experience in modeling the activity of a risk business. During the first part of the twentieth century, they focused on the probability of ruin to assess the stability of their company. In his seminal paper of 1957 Bruno de Finetti criticized this approach and laid the foundations of what would become an increasingly popular topic: the study of dividend strategies. The contributions made by actuaries in that field constitute a substantial body of knowledge, whose interest is relevant not only to insurance but also to a much broader range of areas of practice. In this paper we aim at a taxonomical synthesis of the 50 years of actuarial research that followed de Finetti’s original paper.

Strategies for Dividend Distribution: A Review

In today's world of financial uncertainty, one major public concern is to assess (and possibly improve) the stability of companies that take on risks. Actuaries are aware of that issue for a very long time and have a great experience in modeling the activity of a risk business. During the first part of the twentieth century, they focused on the probability of ruin to assess the stability of their company. In his seminal paper, Bruno de Finetti (1957) criticized this approach and laid the foundations of what would become an increasingly popular topic: the study of dividend strategies. The contributions made by actuaries in that field constitute a substantial body of knowledge, whose interest is relevant not only to insurance, but also to a much broader range of areas of practice. In this paper, we aim at a taxonomical synthesis of the 50 years of actuarial research that followed de Finetti's original paper.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1330638_code1091450.pdf?abstractid=1330638&mirid=1&type=2

In the framework of the classical compound Poisson process in collective risk theory, we study a modification of the horizontal dividend barrier strategy by introducing random observation times at which dividends can be paid and ruin can be observed. This model contains both the continuous-time and the discrete-time risk model as a limit and represents a certain type of bridge between them which still enables the explicit calculation of moments of total discounted dividend payments until ruin. Numerical illustrations for several sets of parameters are given and the effect of random observation times on the performance of the dividend strategy is studied.

Randomized observation periods for the compound Poisson risk model: Dividends

In the framework of collective risk theory, we consider a compound Poisson risk model for the surplus process where the process (and hence ruin) can only be observed at random observation times. For Erlang(n) distributed inter-observation times, explicit expressions for the discounted penalty function at ruin are derived. The resulting model contains both the usual continuous-time and the discrete-time risk model as limiting cases, and can be used as an effective approximation scheme for the latter. Numerical examples are given that illustrate the effect of random observation times on various ruin-related quantities.

/pdf/randomized-observation-periods-for-the-compound-poisson-risk-3354too43b.pdf

Randomized observation periods for the compound Poisson risk model: the discounted penalty function

The structure of various Gerber–Shiu functions in Sparre Andersen models allowing for possible dependence between claim sizes and interclaim times is examined. The penalty function is assumed to depend on some or all of the surplus immediately prior to ruin, the deficit at ruin, the minimum surplus before ruin, and the surplus immediately after the second last claim before ruin. Defective joint and marginal distributions involving these quantities are derived. Many of the properties in the Sparre Andersen model without dependence are seen to hold in the present model as well. A discussion of Lundberg’s fundamental equation and the generalized adjustment coefficient is given, and the connection to a defective renewal equation is considered. The usual Sparre Andersen model without dependence is also discussed, and in particular the case with exponential claim sizes is considered.

Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models

We consider the dual model, which is appropriate for modeling the surplus of companies with deterministic expenses and stochastic gains, such as pharmaceutical, petroleum or commission-based companies. Dividend strategies for this model that can be found in the literature include the barrier strategy (e.g.,  Avanzi et al., 2007 ) and the threshold strategy (e.g.,  Cheung, 2008 ), where dividend decisions are made continuously. While in practice the financial position of a company is typically monitored frequently, dividend decisions are only made periodically along with the publication of its books. In this paper, we introduce a dividend barrier strategy whereby dividend decisions are made only periodically, but still allow ruin to occur at any time (as soon as the surplus is exhausted). This is in contrast to Albrecher et al. (2011a) , who introduced periodic dividend payments in the Cramer–Lundberg surplus model, albeit with periodic ruin opportunities as well. Under the assumption that the time intervals between dividend decisions are Erlang ( n ) distributed, we derive integro-differential equations for the Laplace transform of the time to ruin and the expected present value of dividends until ruin. These are then solved with the help of probabilistic arguments. We also provide a recursive algorithm to compute these quantities. Finally, some numerical studies are presented, which aim at illustrating how our assumptions about dividend payments and ruin occurrence compare with those of the classical barrier strategy.

/pdf/on-a-periodic-dividend-barrier-strategy-in-the-dual-model-2318rwoxns.pdf

On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency

In this paper we consider an extension of the Sparre Andersen insurance risk model by relaxing one of its independence assumptions. The newly proposed dependence structure is introduced through the premise that the joint distribution of the interclaim time and the subsequent claim size is bivariate phase-type (see, e.g. Assaf et al. (1984) and Kulkarni (1989)). Relying on the existing connection between risk processes and fluid flows (see, e.g. Badescu et al. (2005), Badescu, Drekic and Landriault (2007), Ramaswami (2006), and Ahn, Badescu and Ramaswami (2007)), we construct an analytically tractable fluid flow that leads to the analysis of various ruin-related quantities in the aforementioned risk model. Using matrix-analytic methods, we obtain an explicit expression for the Gerber-Shiu discounted penalty function (see Gerber and Shiu (1998)) when the penalty function depends on the deficit at ruin only. Finally, we investigate how some ruin-related quantities involving the surplus immediately prior to ruin can also be analyzed via our fluid flow methodology.

/pdf/dependent-risk-models-with-bivariate-phase-type-w8zf17i48t.pdf

Eric C.K. Cheung

Papers

Randomized observation periods for the compound Poisson risk model: Dividends

Randomized observation periods for the compound Poisson risk model: the discounted penalty function

Structural properties of Gerber-Shiu functions in dependent Sparre Andersen models

On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency

Dependent risk models with bivariate phase-type distributions