A proof of a conjecture in the Cram\'er-Lundberg model with investments

TL;DR

This paper proves a conjecture regarding the ruin probability in the Cramér-Lundberg model with investments, showing conditions under which ruin probability decays algebraically or is certain, depending on the asset's parameters.

Contribution

It establishes a proof for a conjecture about ruin probabilities in the Cramér-Lundberg model with investments, considering the impact of claim size caps and asset parameters.

Findings

Ruin probability decays algebraically if 2a/σ^2 > 1 with claim caps.

Ruin probability is certain for all initial capital if 2a/σ^2 ≤ 1 without claim caps.

The results clarify the impact of investment parameters on ruin risk.

Abstract

In this paper, we discuss the Cram\'er-Lundberg model with investments, where the price of the invested risk asset follows a geometric Brownian motion with drift $a$ and volatility $\sigma> 0.$ By assuming there is a cap on the claim sizes, we prove that the probability of ruin has at least an algebraic decay rate if $2a/\sigma^2 > 1$. More importantly, without this assumption, we show that the probability of ruin is certain for all initial capital $u$, if $2a/\sigma^2 \le 1$.