Distributional equations and the ruin problem for the Sparre Andersen model with investments

TL;DR

The paper investigates ruin probabilities in the Sparre Andersen model with investments by leveraging advanced implicit renewal theory. It recasts ruin tails in terms a stochastic affine equation $Y \stackrel{d}{=} AY+B$ and uses Buraczewski–Damek–Mikosch results to relax earlier conditions, proving power-law decay $\Psi(u) \sim C u^{-\beta}$ under broad moment and support assumptions. A key methodological contribution is the BD-M Prop. 2.5.4-based criterion that guarantees the tail of $Y_{\infty}$ is unbounded by exhibiting affine components that can both exceed and fall below unity. The results cover non-life, annuity, and mixed models, with detailed proofs showing the existence of non-null events for the affine pair $(M_1^k,Q_1^k)$ across variants, thus enabling the desired ruin asymptotics and extending prior work on investments in risky assets.

Abstract

This note is an addendum to the work initiated by Eberlein, Kabanov, and Schmidt and developed further by Kabanov and Promyslov on the asymptotics of the ruin probabilities in the Sparre Andersen model with investments in a risky asset. Using more advanced methods of the implicit renewal theory, we provide complements to some results of the mentioned works.

Theorems & Definitions (15)

• theorem 1
• theorem 2
• theorem 3
• theorem 4: mixed case
• lemma thmcounterlemma: EKS, Lemma 3.1
• theorem 5: see EKS, Theorem 4.2
• proposition thmcounterproposition: BDM, Proposition 2.5.4
• corollary thmcountercorollary
• proposition thmcounterproposition
• proposition thmcounterproposition