Ruin probability for renewal risk models with neutral net profit condition

TL;DR

This paper analyzes ruin probabilities in renewal risk models, especially when the net profit condition is neutral, providing simplified proofs for unavoidable ruin in such cases across various risk models.

Contribution

It offers a simplified argument for unavoidable ruin when claims occur on average as premiums are gained, extending to classical and Sparre Andersen models.

Findings

Unavoidable ruin occurs when claims on average match premiums.

Simplified proof techniques for ruin probability in neutral net profit cases.

Applicability to classical and Sparre Andersen risk models.

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 Cram\'er-Lundberg risk process, and the more general E. Sparre Andersen model.