Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotics of Ruin Probabilities in a Subordinated Cramér-Lundberg Model

Jonathan Klinge, Maren Diane Schmeck

Abstract

We study a dynamic model of a non-life insurance portfolio. The foundation of the model is a compound Poisson process that represents the claims side of the insurer. To introduce clusters of claims appearing, e.g. with catastrophic events, this process is time-changed by a Lévy subordinator. The subordinator is chosen so that it evolves, on average, at the same speed as calendar time, creating a trade-off between intensity and severity. We show that such a transformation always has a negative impact on the probability of ruin. Despite the expected total claim amount remaining invariant, it turns out that the probability of ruin as a function of the initial capital falls arbitrarily slowly depending on the choice of the subordinator.

Asymptotics of Ruin Probabilities in a Subordinated Cramér-Lundberg Model

Abstract

We study a dynamic model of a non-life insurance portfolio. The foundation of the model is a compound Poisson process that represents the claims side of the insurer. To introduce clusters of claims appearing, e.g. with catastrophic events, this process is time-changed by a Lévy subordinator. The subordinator is chosen so that it evolves, on average, at the same speed as calendar time, creating a trade-off between intensity and severity. We show that such a transformation always has a negative impact on the probability of ruin. Despite the expected total claim amount remaining invariant, it turns out that the probability of ruin as a function of the initial capital falls arbitrarily slowly depending on the choice of the subordinator.
Paper Structure (11 sections, 10 theorems, 78 equations, 3 figures)

This paper contains 11 sections, 10 theorems, 78 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Subordinated Compound Poisson Process
  3. Model
  4. Stochastic Dominance of Subordinated Jump Sizes
  5. Heavy- and Light-Tailedness of Z
  6. Adjustment Coefficient in the Subordinated Model: Light-Tailed Jump Sizes
  7. Asymptotic Ruin Probabilities: Heavy-Tailed Jump Sizes
  8. Preliminary Definitions
  9. Subexponential Initial Jump Size
  10. Regularly Varying Subordinator
  11. Concluding Remarks

Key Result

Theorem 1

The process $(Y_t)_{t\geq 0}:=(C_{\Lambda_t})_{t\geq 0}$, defined in (defsubordinatedprocess), has a compound Poisson representation with intensity $\psi_{\Lambda}(\lambda)$ for the Laplace exponent $\psi_{\Lambda}$ of $\Lambda$ and jump size distribution where $\nu_Y$ is the Lévy measure of $Y$, which is given by

Figures (3)

  • Figure 1: Trajectory of a subordinated compound Poisson process, subordinated by drifted compound Poisson process
  • Figure 2: Jump size distribution after subordination
  • Figure 3: The adjustment function for different subordinators

Theorems & Definitions (25)

  • Theorem 1
  • proof
  • Example 1
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Definition 1
  • Lemma 2
  • proof
  • ...and 15 more