Parisian ruin of locally self-similar Gaussian processes

TL;DR

This paper derives precise tail asymptotics for the Parisian ruin probability in Gaussian risk models driven by locally self-similar processes, extending classical models with explicit asymptotic formulas.

Contribution

It introduces a novel analysis of Parisian ruin probabilities for non-stationary Gaussian processes with self-similarity, including new asymptotic representations involving Pickands-type constants.

Findings

Asymptotic ruin probabilities depend on local variance decay, self-similarity index, and trend exponent.

Explicit asymptotic formulas involve Parisian Pickands-type constants.

The method extends existing techniques to locally self-similar Gaussian risk models.

Abstract

We derive exact tail asymptotics of the Parisian ruin probability for Gaussian risk models driven by locally self-similar Gaussian processes with a power-type deterministic trend. The considered setting includes non-stationary Gaussian processes whose local correlation structure is governed by a self-similar limiting process, extending classical fractional Brownian motion models. The asymptotic behaviour is shown to depend on the interplay between the local variance decay, the self-similarity index, and the trend exponent, leading to several distinct regimes. In each regime, the ruin probability admits an explicit asymptotic representation involving Parisian Pickands-type constants. The analysis relies on a uniform Pickands lemma allowing for families of limiting Gaussian fields, extending existing double-sum techniques and enabling the treatment of locally self-similar Gaussian risk models.