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2023 | Buch

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The Cramér–Lundberg Model and Its Variants

A Queueing Perspective

verfasst von: Michel Mandjes, Onno Boxma

Verlag: Springer Nature Switzerland

Buchreihe : Springer Actuarial

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book offers a comprehensive examination of the Cramér–Lundberg model, which is the most extensively researched model in ruin theory. It covers the fundamental dynamics of an insurance company's surplus level in great detail, presenting a thorough analysis of the ruin probability and related measures for both the standard model and its variants.
Providing a systematic and self-contained approach to evaluate the crucial quantities found in the Cramér–Lundberg model, the book makes use of connections with related queueing models when appropriate, and its emphasis on clean transform-based techniques sets it apart from other works. In addition to consolidating a wealth of existing results, the book also derives several new outcomes using the same methodology.
This material is complemented by a thoughtfully chosen collection of exercises. The book's primary target audience is master's and starting PhD students in applied mathematics, operations research, and actuarial science, although it also serves as a useful methodological resource for more advanced researchers. The material is self-contained, requiring only a basic grounding in probability theory and some knowledge of transform techniques.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Cramér-Lundberg Model

Abstract

In this chapter we discuss the conventional Cramér-Lundberg ruin model. The focus lies on evaluating transforms related to the all-time and time-dependent ruin probabilities (where the latter, in our setting, concerns ruin before an exponentially distributed amount of time). An important role is played by a duality with the M/G/1 queueing model. We present four independent analysis techniques; they differ in the sense whether the ruin model or the corresponding queueing model (or a mixture of both) has been used.

Michel Mandjes, Onno Boxma

Chapter 2. Asymptotics

Abstract

This chapter provides asymptotic expressions for the ruin probability in the regime that the initial reserve level u grows large. One needs to distinguish between the case that the claim sizes are light-tailed, in which the ruin probability decays essentially exponentially, and the case that the claim sizes are heavy-tailed, in which the ruin probability decays as the complementary distribution function of a residual claim size. While the focus in this chapter is on the asymptotics of the all-time ruin probability \(p(u)\), we also briefly discuss the asymptotics of its finite-time counterpart \(p(u,t).\) We conclude this chapter by some comments on another limiting regime, corresponding to the heavy-traffic scaling in queueing theory.

Michel Mandjes, Onno Boxma

Chapter 3. Regime Switching

Abstract

This chapter focuses on the evaluation of the transform of the ruin probability in a regime-switching (or Markov modulated) version of the standard Cramér-Lundberg model. In the derivation various elements from Chap. 1 are relied upon, in particular the idea of conditioning on the first event and the Wiener-Hopf decomposition. The results established in this chapter also facilitate the analysis of the conventional (non-modulated, that is) Cramér-Lundberg model over a phase-type horizon (rather than an exponentially distributed horizon). Finally, we comment on a setup in which the model parameters are periodically resampled, and argue that it fits in the modelling framework discussed in the chapter.

Michel Mandjes, Onno Boxma

Chapter 4. Interest and Two-Sided Jumps

Abstract

In this chapter we present an in-depth analysis of a generalization of the Cramér-Lundberg model. Three distinguishing additional elements have been incorporated: (1) the insurance firm receives interest over its surplus level, (2) besides claims, leading to negative jumps of the surplus level process, we also allow positive jumps, and (3) as before we obtain the probability of ruin (transformed with respect to the initial capital surplus level) before an exponentially distributed time, but now jointly with three other quantities: the corresponding time of ruin, the undershoot, and the overshoot.

Michel Mandjes, Onno Boxma

Chapter 5. Threshold-Based Net Cumulative Claim Process

Abstract

This chapter discusses a variant of the Cramér-Lundberg model in which the net cumulative claim process obeys different stochastic dynamics (in terms of the claim arrival rate, premium rate, and claim-size distribution) above and below a threshold v. For this setting of a threshold-based net cumulative claim process we evaluate the ruin probability over an exponentially distributed interval. An important role is played by the concept of scale functions.

Michel Mandjes, Onno Boxma

Chapter 6. Level-Dependent Dynamics

Abstract

This chapter considers a variant of the Cramér-Lundberg model which has the special feature that the claim arrival rate \(\lambda (x)\) and the premium rate \(r(x)\) are functions of the current surplus level x. If only the premium rate depends on that surplus level, then duality holds with a so-called shot-noise M/G/1 queue, which can be used to derive the ruin probability from a known queueing result. When both rates are level-dependent, we exploit the fact that the integro-differential equation for the survival probability only involves those two rates as a fraction \(r(x)/\lambda (x)\). This reduces the determination of the ruin probability to that for the case in which only the premium rate is level-dependent. Then we consider a model in which the claim interarrival times depend on the current surplus level in a specific way: they equal an exponentially distributed quantity minus a fraction of the current surplus level, truncated at 0. The chapter is concluded by the analysis of a model in which tax payments are deducted from the premium income whenever the surplus process is at a running maximum, leading to the so-called tax identity.

Michel Mandjes, Onno Boxma

Chapter 7. Multivariate Ruin

Abstract

In this chapter we focus on a multivariate variant of the conventional Cramér-Lundberg model. Imposing an ordering condition on the individual net cumulative claim processes, it turns out that the distribution of the joint running maximum can be derived, which can be used to evaluate ruin probabilities in a multivariate context. We start by analyzing the bivariate case, to then extend the reasoning to the higher-dimensional setting. The method relies upon the Kolmogorov forward equations underlying the associated queueing process. The solution reveals a so-called quasi-product form structure. We also point out how the results from this section can be translated into corresponding results for tandem queueing networks. We conclude the chapter by deriving the corresponding multivariate Gerber-Shiu metrics (including ruin times, undershoots, and overshoots).

Michel Mandjes, Onno Boxma

Chapter 8. Arrival Processes with Clustering

Abstract

In this chapter we focus on the Cramér-Lundberg model driven by a claim arrival process with randomly fluctuating rate. We consecutively discuss models in which the arrival rate evolves as an M/G/\(\infty \) queue (to do justice to a fluctuating number of customers), as a shot-noise process (to model the impact of catastrophic events) and as a Hawkes process (to model the effect of claims triggering additional claims). The main objective is to determine, in the light-tailed context, the decay rate of the ruin probability. The proofs rely either on applying a change-of-measure, or on a large deviations based argumentation.

Michel Mandjes, Onno Boxma

Chapter 9. Dependence Between Claim Sizes and Interarrival Times

Abstract

Where in the conventional Cramér-Lundberg model claim sizes and claim interarrival times are independent, this chapter studies the case in which this independence assumption is lifted. We first consider two models in which the claim size is correlated with the previous claim interarrival time, and subsequently two models in which the claim interarrival time is correlated with the previous claim size. In all models the double transform pertaining to the time-dependent ruin probability is computed.

Michel Mandjes, Onno Boxma

Chapter 10. Advanced Bankruptcy Concepts

Abstract

So far we focused on the event of ruin, corresponding to the reserve process dropping below 0. This chapter studies various bankruptcy concepts, in which, besides the reserve level becoming negative, an additional condition has to be fulfilled. In the first variant, the net reserve process is inspected at Poisson instants, and bankruptcy occurs if the reserve level is below zero at such an inspection time. For this setting an adapted version of the Pollaczek-Khinchine theorem is derived, as well as an appealing decomposition. In the second variant there is bankruptcy if the reserve process is uninterruptedly below 0 for a sufficiently long time, whereas in the third variant the bankruptcy criterion is based on the total time with a negative surplus.

Michel Mandjes, Onno Boxma

Backmatter

Titel: The Cramér–Lundberg Model and Its Variants
verfasst von: Michel Mandjes
Onno Boxma
Verlag: Springer Nature Switzerland
Electronic ISBN: 978-3-031-39105-7
Print ISBN: 978-3-031-39104-0
DOI: https://doi.org/10.1007/978-3-031-39105-7