Parisian ruin with power-asymmetric variance near the optimal point with application to many-inputs proportional reinsurance
Pavel Ievlev
TL;DR
The paper addresses the exact asymptotics of the Parisian ruin probability for Gaussian processes with power asymmetric variance near a unique optimal point, extending previous results to Pickands scale and applying the findings to many-inputs proportional reinsurance with fractional Brownian motion. The authors develop a framework that uses uniform local Pickands lemmas and Pickands/Piterbarg/Talagrand regimes to derive precise asymptotics, expressed through generalized Pickands constants and related limits, and they provide a corollary for the multivariate reinsurance model. The approach combines a careful decomposition into vicinities and intervals, continuity properties of the Parisian functional, and auxiliary lemmas to control contributions away from the optimal point. The results advance the theory of Gaussian extremes in the Parisian setting and offer practical ruin-probability estimates for multivariate long-range dependent risk processes, improving understanding of risk in systems with many inputs. Overall, the work broadens the applicability of Parisian ruin analysis to high-dimensional, dependence-rich insurance models and clarifies the role of variance asymmetry and scaling in determining the limiting ruin behavior.
Abstract
This paper investigates the Parisian ruin probability for processes with power-asymmetric behavior of the variance near the unique optimal point. We derive the exact asymptotics as the ruin boundary tends to infinity and extend the previous result arXiv:1504.07061 to the case when the length of Parisian interval is of Pickands scale. As a primary application, we extend the recent result arXiv:2010.00222 on the many inputs proportional reinsurance fractional Brownian motion risk model to the Parisian ruin.