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This monograph presents a time-dynamic model for multivariate claim counts in actuarial applications.

Inspired by real-world claim arrivals, the model balances interesting stylized facts (such as dependence across the components, over-dispersion and the clustering of claims) with a high level of mathematical tractability (including estimation, sampling and convergence results for large portfolios) and can thus be applied in various contexts (such as risk management and pricing of (re-)insurance contracts). The authors provide a detailed analysis of the proposed probabilistic model, discussing its relation to the existing literature, its statistical properties, different estimation strategies as well as possible applications and extensions.

Actuaries and researchers working in risk management and premium pricing will find this book particularly interesting. Graduate-level probability theory, stochastic analysis and statistics are required.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Motivation and Model

Abstract
A classical task in actuarial risk management and pricing of (re-)insurance policies is to model the number and severity of insurance claims over a given time period. Concerning the number of claims—which is the primary object of our study—the simplest and most traditional approach is to resort to Poisson-distributed random variables, see Eq. 5.1) in the Appendix, modelling the number of claims occurring in a given time period. Popular extensions of this classical approach include more general distributions derived from the Poisson law (e.g. by randomizing the parameter) and multivariate treatments with dependent random variables for different loss categories. Still, this remains a static approach in the sense that extrapolations and interpolations to other time periods are not obvious and typically require assumptions on a dependence structure in time.
Daniela Anna Selch, Matthias Scherer

Chapter 2. Properties of the Model

Abstract
For the application and estimation of the claim-count process it is essential to fully understand how the model behaves. Section 2.1 focuses on the finite-dimensional distribution of the process and develops formulas for the probability mass function and many related quantities. In Sect. 2.2 the Lévy characteristics of the time-changed model are derived, which directly lead to a second stochastic representation of the process as a multivariate Poisson cluster process.
Daniela Anna Selch, Matthias Scherer

Chapter 3. Estimation of the Parameters

Abstract
This chapter treats the problem of estimating the model parameters from claim count data. In general, statistical inference for multivariate processes is far from trivial and often computationally expensive. In Sect. 3.1 four estimation procedures are developed which essentially try to fit a multivariate distribution, either the infinitely divisible process distribution or the jump size distribution (and jump intensity).
Daniela Anna Selch, Matthias Scherer

Chapter 4. Applications and Extensions

Abstract
Actuarial applications of the presented model are examined in Sect. 4.1. For this purpose, iid claim sizes are introduced and premium principles and other actuarial risk measures are evaluated. The results are compared to those stemming from a claim number process with independent Poisson marginals and, in addition, to those of a claim number process with the same Poisson cluster marginals as the proposed model, but independence between the components.
Daniela Anna Selch, Matthias Scherer

Chapter 5. Appendix: Technical Background

Abstract
The standard process to model claim arrivals in actuarial science is the Poisson process and its generalizations.
Daniela Anna Selch, Matthias Scherer

Backmatter

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