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.
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- 1.
These assumptions may be too restrictive for certain applications, for instance, if contagion effects are observed, as may be the case in motor insurance policies where an accident caused by a policyholder increases the likelihood of future accidents, or if a seasonal effect influences claim arrivals that originate from certain types of natural catastrophes. When modelling motor insurance claim numbers, [40, Sect. 2.9, p. 90] argue that independence can be assumed between different policyholders, though assuming independence over time for a single policyholder seems questionable. Chapter 4 sketches some extensions of the model which incorporate such properties by relaxing the Lévy property.
- 2.
The negative binomial distribution is a popular alternative to the Poisson distribution under the presence of overdispersion and is well studied. Some details and many references can be found in [73, Chap. 5, p. 199]. An example comparing the shape of the probability mass function to the Poisson distribution and other mixed Poisson distributions can be seen in Fig. 2.1.
- 3.
This distribution is a special two parameter subclass of the three parameter Sichel distribution. Some alternative parametrizations and a recursive formula for the probability mass function are referenced in [73, Chap. 15, p. 455]. An example the shape of the probability mass function can again be found in Fig. 2.1.
- 4.
Here we consider higher order partial derivatives \(\varphi _{\varvec{\varLambda }_{\varvec{t}}}^{(\varvec{k})}(\varvec{\lambda }):= \frac{\partial ^{|\varvec{k}|}\varphi _{\varvec{\varLambda }_{\varvec{t}}}(\varvec{\lambda })}{\partial ^{k_1}\lambda _1\cdots \partial ^{k_d}\lambda _d}(\varvec{\lambda })\).
- 5.
For example, the worst obtainable value at risk (VaR) for a portfolio with given marginal distributions does not necessarily correspond to the highest possible correlation. This stems from the fact that the VaR is not subadditive and situations may exist where the VaR of the sum even exceeds the sum of VaRs of the individuals. Much research regarding bounds for the distribution of a sum with given marginals but unspecified dependence structure has been carried out.
- 6.
The ties in the calculation of Spearman’s rho are accounted for by averaging the values of the affected ranks. For Kendall’s tau, the tie-adjusted version tau-b is calculated. For details, we refer to the MATLAB® documentation and the references given there.
- 7.
Here we are looking at the static distribution of the process at a fixed point in time. We will see in the next section, however, that the subordinator choice has a pronounced impact on the jump size distribution of the process and, hence, determines how the process evolves over time.
- 8.
We use a dot to indicate that a space is punctured at the origin \(\dot{\mathbb {R}}^d_{\ge 0}:=\mathbb {R}^d_{\ge 0}\setminus \{\varvec{0}\}\), \(\varvec{0} \in \mathbb {R}^d\) the zero vector. Also, \(\mathscr {B}(\cdot )\) denotes the Borel \(\sigma \)-algebra and \(\Vert \cdot \Vert \) the Euclidean norm.
- 9.
If \(\varLambda _t = at\) for some \(a>0\), then the components \(L_t^1\) and \(L_t^2\) remain independent and a.s. never jump together.
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Selch, D.A., Scherer, M. (2018). Properties of the Model. In: A Multivariate Claim Count Model for Applications in Insurance. Springer Actuarial. Springer, Cham. https://doi.org/10.1007/978-3-319-92868-5_2
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DOI: https://doi.org/10.1007/978-3-319-92868-5_2
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