[Paper Review] On total claim amount for marked Poisson cluster models

Bojan Basrak, Olivier Wintenberger+1 more|arXiv (Cornell University)|May 9, 2018

Probability and Risk ModelsDecision Sciences11 references3 citations

TL;DR

This paper establishes limit theorems for the total claim amount in marked Poisson cluster models, where claim arrivals exhibit clustering and marks influence both claim size and future arrival rates. It proves that under regular variation of claim sizes, the normalized total claim amount converges in distribution to a stable law with infinite variance when claim size tails are heavy, or to a normal distribution under second moment conditions.

ABSTRACT

We study the asymptotic distribution of the total claim amount for marked Poisson cluster models. The marks determine the size and other characteristics of the individual claims and potentially influence arrival rate of the future claims. We find sufficient conditions under which the total claim amount satisfies the central limit theorem or alternatively tends in distribution to an infinite variance stable random variable. We discuss several Poisson cluster models in detail, paying special attention to the marked Hawkes processes as our key example.

Motivation & Objective

• To study the asymptotic distribution of the total claim amount in marked Poisson cluster processes, extending classical Cramér–Lundberg models to include clustering effects.
• To determine conditions under which the total claim amount satisfies a central limit theorem or converges to a stable law with infinite variance.
• To analyze the impact of dependence between marks and claim arrival times on the limiting behavior of the total claim amount.
• To provide a detailed analysis of marked Hawkes processes as a key example of such cluster models.
• To establish functional limit theorems for sums of regularly varying i.i.d. random variables subordinated to an independent renewal process, applicable to claim size processes.

Proposed method

• Models claim arrivals as a marked Poisson point process on [0,∞) × S, where marks in a metric space S determine claim size and influence future arrivals.
• Defines the total claim amount S(t) as the integral of claim size function f(a) over the point process N on [0,t] × S.
• Uses a cluster structure where each original point Γi generates an independent cluster process G^Ai with distribution K(Ai, ⋅), allowing marks to affect cluster intensity.
• Applies functional limit theorems for sums of i.i.d. regularly varying random variables subordinated to an independent renewal process to derive the asymptotic behavior of S(t).
• Employs regular variation theory and Tauberian theorems to characterize the tail behavior of the total claim amount under heavy-tailed claim size distributions.
• Applies the results to marked Hawkes processes, showing convergence to stable laws when claim size tails are regularly varying with index ∈ (0,2).

Experimental results

Research questions

• RQ1 Under what conditions does the total claim amount in a marked Poisson cluster model satisfy a central limit theorem?
• RQ2 When do the claim sizes have infinite variance, and what is the limiting distribution of the normalized total claim amount?
• RQ3 How does the dependence between marks and arrival times affect the asymptotic behavior of the total claim amount?
• RQ4 What is the precise asymptotic distribution of the total claim amount in marked Hawkes processes with regularly varying claim sizes?
• RQ5 Can the functional limit theorem for subordinated sums be applied to derive limit laws for the total claim amount in cluster models?

Key findings

• When claim sizes have finite variance and second moment conditions are satisfied, the normalized total claim amount S(t) converges in distribution to a normal (Gaussian) random variable.
• When claim sizes are regularly varying with tail index α ∈ (0,1), the normalized total claim amount S(t)/a_⌊νt⌋ converges in distribution to a strictly stable α-stable random variable Gα.
• For α ∈ (1,2), the centered total claim amount (S(t) − tνμD)/a_⌊νt⌋ converges in distribution to a strictly stable α-stable random variable Gα, provided the expected number of delayed claims decays sufficiently fast.
• The convergence to stable laws holds under moment conditions involving the tail behavior of the cluster size distribution and the decay rate of the expected number of claims arriving after time t.
• The results are extended to the stationary version of the marked Hawkes process, confirming the same limit behavior under identical conditions.
• The paper establishes that the error term εt from incomplete claims is asymptotically negligible (oP(at)) under the stated decay conditions on the tail of the cluster size distribution.