On the total claim amount for marked Poisson cluster models

Basrak, Bojan; Wintenberger, Olivier; Žugec, Petra

doi:10.1017/apr.2019.15

Cited by 7 publications

(10 citation statements)

References 25 publications

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“…Associated marked renewal cluster model is then called marked Poisson cluster process, see [12], cf. [6].…”

Section: Random Maximamentioning

confidence: 99%

On extremes of random clusters and marked renewal cluster processes

Basrak¹,

Milinčević²,

Žugec³

2022

Preprint

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The article describes the limiting distribution of the extremes of observations that arrive in clusters. We start by studying the tail behaviour of an individual cluster and then we apply the developed theory to determine the limiting distribution of max{Xj : j = 0, . . . , K(t)}, where K(t) is the number of i.i.d. observations (Xj ) arriving up to the time t according to a general marked renewal cluster process. The results are illustrated in the context of some commonly used Poisson cluster models such as the marked Hawkes process.

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“…Associated marked renewal cluster model is then called marked Poisson cluster process, see [12], cf. [6].…”

Section: Random Maximamentioning

confidence: 99%

On extremes of random clusters and marked renewal cluster processes

Basrak¹,

Milinčević²,

Žugec³

2022

Preprint

Self Cite

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“…The process X in (1.1) can be used to model the total claims in insurance, where A ( t ) is the arrival process of cluster claims, the are the cluster sizes, the variables are the claim sizes for each cluster, and are the delays for the claims to arrive in each cluster. See, for example, the book by Daley and Vere-Jones [4] and the recent work in [1] and references therein. When the arrival process A is Poisson, under the i.i.d.…”

Section: Examplesmentioning

confidence: 99%

“…conditions on the claim sizes and delays, the distribution of X can be characterized using the probability generating or characteristic functionals [4, 31]. In [1, 32], central limit theorems with Gaussian and infinite-variance stable limits are proved and used to approximate the total claim distributions as . Our results provide distributional approximations for the total claim size at each time t when the arrival rate of cluster claims is large; these approximations are valid for any general non-stationary arrival processes, as well as for various scenarios of correlated claims and delays discussed above.…”

Section: Examplesmentioning

confidence: 99%

“…Shot noise processes with cluster arrivals and various forms of the shot shape functions have been studied in [1, 7, 31, 32]. In these papers, people have investigated the general formula for the characteristic functional [7, 31], long-range dependence under certain structural conditions on the response function [31], and central limit theorem results with normal limit distribution [32] and with stable limit distribution [1]. Similar results for shot noise processes without cluster arrivals were established in [9, 20, 21, 33, 34].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Shot noise processes with randomly delayed cluster arrivals and dependent noises in the large-intensity regime

Li¹,

Pang²

2021

Adv. Appl. Probab.

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We study shot noise processes with cluster arrivals, in which entities in each cluster may experience random delays (possibly correlated), and noises within each cluster may be correlated. We prove functional limit theorems for the process in the large-intensity asymptotic regime, where the arrival rate gets large while the shot shape function, cluster sizes, delays, and noises are unscaled. In the functional central limit theorem, the limit process is a continuous Gaussian process (assuming the arrival process satisfies a functional central limit theorem with a Brownian motion limit). We discuss the impact of the dependence among the random delays and among the noises within each cluster using several examples of dependent structures. We also study infinite-server queues with cluster/batch arrivals where customers in each batch may experience random delays before receiving service, with similar dependence structures.

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“…Generally, times of events together with accompanying measures can be analyzed as marked point processes, e.g., in case of ultrahigh-frequency data, see Engle (2000). Stochastic methods for modeling the total claim amount via marked Poisson cluster models have been recently proposed by Basrak et al (2018). Here, the marks can take values only in a finite-dimensional space and have a common distribution.…”

Section: Current Statusmentioning

confidence: 99%

Infinitely Stochastic Micro Forecasting

Maciak¹,

Okhrin²,

Pešta³

2019

Preprint

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Forecasting costs is now a front burner in empirical economics. We propose an unconventional tool for stochastic prediction of future expenses based on the individual (micro) developments of recorded events. Consider a firm, enterprise, institution, or state, which possesses knowledge about particular historical events. For each event, there is a series of several related subevents: payments or losses spread over time, which all leads to an infinitely stochastic process at the end. Nevertheless, the issue is that some already occurred events do not have to be necessarily reported. The aim lies in forecasting future subevent flows coming from already reported, occurred but not reported, and yet not occurred events. Our methodology is illustrated on quantitative risk assessment, however, it can be applied to other areas such as startups, epidemics, war damages, advertising and commercials, digital payments, or drug prescription as manifested in the paper. As a theoretical contribution, inference for infinitely stochastic processes is developed. In particular, a non-homogeneous Poisson process with non-homogeneous Poisson processes as marks is used, which includes for instance the Cox process as a special case.

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On the total claim amount for marked Poisson cluster models

Cited by 7 publications

References 25 publications

On extremes of random clusters and marked renewal cluster processes

On extremes of random clusters and marked renewal cluster processes

Shot noise processes with randomly delayed cluster arrivals and dependent noises in the large-intensity regime

Infinitely Stochastic Micro Forecasting

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