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Risk theory in a finite customer-pool setting

Michel Mandjes, Daniël Rutgers

Abstract

This paper investigates an insurance model with a finite number of major clients and a large number of small clients, where the dynamics of the latter group are modeled by a spectrally positive Lévy process. We begin by analyzing this general model, in which the inter-arrival times are exponentially distributed (though not identically), and derive the closed-form Laplace transform of the ruin probability. Next, we examine a simplified version of the model involving only the major clients, and explore the tail asymptotics of the ruin probability, focusing on the cases where the claim sizes follow phase-type or regularly varying distributions. Finally, we derive the distribution of the overshoot over an exponentially distributed initial reserve, expressed in terms of its Laplace-Stieltjes transform.

Risk theory in a finite customer-pool setting

Abstract

This paper investigates an insurance model with a finite number of major clients and a large number of small clients, where the dynamics of the latter group are modeled by a spectrally positive Lévy process. We begin by analyzing this general model, in which the inter-arrival times are exponentially distributed (though not identically), and derive the closed-form Laplace transform of the ruin probability. Next, we examine a simplified version of the model involving only the major clients, and explore the tail asymptotics of the ruin probability, focusing on the cases where the claim sizes follow phase-type or regularly varying distributions. Finally, we derive the distribution of the overshoot over an exponentially distributed initial reserve, expressed in terms of its Laplace-Stieltjes transform.
Paper Structure (21 sections, 20 theorems, 112 equations, 5 figures)

This paper contains 21 sections, 20 theorems, 112 equations, 5 figures.

Table of Contents

  1. I. Introduction
  2. II. Model & objectives
  3. III. Laplace-Stieltjes transform of running maximum
  4. IV. Phase-type claims
  5. V. Regularly varying claims
  6. VI. Overshoot analysis
  7. VII. Discussion and concluding remarks
  8. Software and data availability
  9. Acknowledgments
  10. CRediT authorship contribution statement
  11. Declaration of competing interest
  12. Funding Information
  13. Declaration of Generative AI and AI-assisted technologies in the writing process

Key Result

Lemma 1

For any $\alpha\geqslant0$ and for any non-negative random variable $Y$,

Figures (5)

  • Figure 1: An example of an instance of a single path in the model with Brownian motions with positive drifts, for $m=5$.
  • Figure 2: Mean (top) and variance (bottom) of the running maximum, as functions of time, in the model with only positive drifts, for different values of $m$. The gray lines denote Monte Carlo based estimated values and are plotted for comparison.
  • Figure 3: Mean (top) and variance (bottom) of the running maximum, as functions of time, in the model with Brownian motions with positive drifts, for different values of $m$. The dotted lines denote the model without the added Brownian motions.
  • Figure 4: Exact, asymptotic and Monte Carlo based estimated values of the ruin probability, for $m=5$ and Erlang distributed claim sizes, in the model with only positive drifts.
  • Figure 5: Laplace inverted, asymptotic and Monte Carlo based estimated values of the ruin probability, for $m=5$ and regular varying distributed claim sizes, in the model with only positive drifts.

Theorems & Definitions (35)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Corollary 1
  • Theorem 1
  • Lemma 3
  • proof
  • Theorem 2
  • proof
  • ...and 25 more