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Ruin probability for renewal risk models with neutral net profit condition

Andrius Grigutis, Arvydas Karbonskis, Jonas Šiaulys

TL;DR

The paper analyzes ruin probabilities in renewal risk processes under the neutral net profit condition $\mathbb{E}X=c\mathbb{E}\theta$ and shows that, in several standard settings, ruin is certain for every nonnegative initial surplus: the discrete-time with periodic $X_i$, the classical Poisson-arrival model, and E. Sparre Andersen’s renewal framework. The authors introduce a simple perturbation $X^*$ with $\mathbb{P}(X^*\le X)=1$ and $\mathbb{E}X^*<c\mathbb{E}\theta$, then leverage the Pollaczek–Khinchine formula (and its variants) to bound the survival probability and pass to the limit as the perturbation vanishes, yielding $\psi(u)=1$ for all $u$. This approach provides a streamlined, accessible alternative to deeper classical results, clarifying how neutrality drives inevitable ruin in these renewal risk processes. The work thus connects renewal theory, PK-type analyses, and ruin criteria to furnish succinct proofs across multiple risk-process paradigms.

Abstract

In ruin theory, the net profit condition intuitively means that the incurred random claims on average do not occur more often than premiums are gained. The breach of the net profit condition causes guaranteed ruin in few but simple cases when both the claims' inter-occurrence time and random claims are degenerate. In this work, we give a simplified argumentation for the unavoidable ruin when the incurred claims on average occur equally as the premiums are gained. We study the discrete-time risk model with $N\in\mathbb{N}$ periodically occurring independent distributions, the classical risk model, also known as the Cramér-Lundberg risk process, and the more general E. Sparre Andersen model.

Ruin probability for renewal risk models with neutral net profit condition

TL;DR

The paper analyzes ruin probabilities in renewal risk processes under the neutral net profit condition and shows that, in several standard settings, ruin is certain for every nonnegative initial surplus: the discrete-time with periodic , the classical Poisson-arrival model, and E. Sparre Andersen’s renewal framework. The authors introduce a simple perturbation with and , then leverage the Pollaczek–Khinchine formula (and its variants) to bound the survival probability and pass to the limit as the perturbation vanishes, yielding for all . This approach provides a streamlined, accessible alternative to deeper classical results, clarifying how neutrality drives inevitable ruin in these renewal risk processes. The work thus connects renewal theory, PK-type analyses, and ruin criteria to furnish succinct proofs across multiple risk-process paradigms.

Abstract

In ruin theory, the net profit condition intuitively means that the incurred random claims on average do not occur more often than premiums are gained. The breach of the net profit condition causes guaranteed ruin in few but simple cases when both the claims' inter-occurrence time and random claims are degenerate. In this work, we give a simplified argumentation for the unavoidable ruin when the incurred claims on average occur equally as the premiums are gained. We study the discrete-time risk model with periodically occurring independent distributions, the classical risk model, also known as the Cramér-Lundberg risk process, and the more general E. Sparre Andersen model.
Paper Structure (5 sections, 4 theorems, 54 equations)

This paper contains 5 sections, 4 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. One auxiliary statement
  3. Discrete-time risk model
  4. Classical risk model
  5. E.S. Andersen's model

Key Result

Theorem 1

Suppose the insurer's surplus process $W(t)$ varies according to the discrete-time risk model discrete_time with $N$ periodically occurring independent discrete and integer-valued non-negative r.v.s $X_i\mathop{=}\limits^{d}X_{i+N}$ and $\theta\equiv 1$. Let $S_N=X_1+X_2+\ldots+X_N$. If the net prof

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • proof
  • proof : Proof of Theorem \ref{['thm:discrete']}
  • proof : Proof of Theorem \ref{['thm:classical']}
  • proof : Proof of Theorem \ref{['thm:Andersen']}