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Erschienen in:
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2018 | OriginalPaper | Buchkapitel

1. Motivation and Model

verfasst von : Daniela Anna Selch, Matthias Scherer

Erschienen in: A Multivariate Claim Count Model for Applications in Insurance

Verlag: Springer International Publishing

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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.

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Nächstes Kapitel Properties of the Model
Fußnoten
1
All limitations of the univariate Poisson model apply to the multivariate version as well. The dispersion observed in claim count data often exceeds that predicted by the Poisson distribution, which is one reason why extensions like mixed Poisson and Cox processes have gained in popularity. Furthermore, claims produced from a univariate homogeneous Poisson process always arrive one after another, while in many real-world applications claims arrive simultaneously. This extends to the multivariate setup as well. Due to independence of the marginal processes, simultaneous claim arrivals are a.s. impossible for each component as well as between different components. Thus, the simple multivariate Poisson process with independent marginals fails to account for cluster arrivals and for dependence between claims arriving in different portfolios.
 
2
The directing processes are all driven by the Lévy subordinator \(\Lambda \), but each one of them is scaled by the component-specific intensity \(\lambda _i\). That is, the directing processes are \(\lambda _i\Lambda \), \(i \in \{1, \ldots , d\}\).
 
3
Multi-index notation is used, i.e. for any \(\varvec{k} \in \mathbb {N}_0^d\), \(\varvec{x} \in \mathbb {R}^d\) we write \(\varvec{x}^{\varvec{k}}:= x_1^{k_1}\dots x_d^{k_d}\), \(|\varvec{x}|:=x_1+\cdots +x_d\), \(\varvec{k}! := k_1! \cdots k_d!\), and for the multinomial coefficient:
$$\begin{aligned} \left( {\begin{array}{c}|\varvec{k}|\\ \varvec{k}\end{array}}\right) := \frac{|\varvec{k}|!}{\varvec{k}!}=\frac{(k_1 + \cdots + k_d)!}{k_1! \cdots k_d!}. \end{aligned}$$
Also, \(\mathbb {N}:=\{1, 2, \ldots \}\) and \(\mathbb {N}_0:=\mathbb {N}\cup \{0\}\).
 
4
Except for the degenerate case \(\Lambda _t = b t\), where the time-changed model consists again of homogeneous independent Poisson processes.
 
5
A popular example is given by shot-noise intensities \(\lambda _t := \sum _{j = 1}^{N_t} X_jf(t-T_j), t\ge 0\). The arrival times \(T_j\), \(j \in \mathbb {N}\), are commonly specified via a second Poisson process, the sequence \(X_j\), \(j\in \mathbb {N}\), consists of iid random variables independent of these arrival times, and f is a deterministic function with \(f(t) = 0\) for \(t<0\). In many applications \(f(t) := \exp \{-\alpha t\}\) for \(\alpha >0\), i.e. the intensity jumps at times \(T_j\) with magnitudes \(X_j\) and the effects wear off with exponential decay rate \(\alpha \). Reference [37] propose shot-noise-driven Cox processes as claim arrival processes for catastrophic events; the resulting risk process and ruin probabilities are investigated in [2]. Reference [97] examine log-normal intensity processes and also consider a bivariate extension. Actuarial applications, in particular pricing of stop-loss contracts, for shot-noise-driven Cox processes are studied in [9]. Reference [64] propose a unifying approach, where the intensity process is defined as an integral w.r.t. a Lévy basis and [35] focus on affine intensity processes, in particular Feller processes.
 
Metadaten
Titel
Motivation and Model
verfasst von
Daniela Anna Selch
Matthias Scherer
Copyright-Jahr
2018
Buch
A Multivariate Claim Count Model for Applications in Insurance

Print ISBN: 978-3-319-92867-8

Electronic ISBN: 978-3-319-92868-5

Copyright-Jahr: 2018

DOI
https://doi.org/10.1007/978-3-319-92868-5_1