Extremes of uncorrelated gamma-reflected Brownian motions with trend
Abstract
This paper investigates the asymptotic behavior of the simultaneous ruin probability for a system of $n$ independent $\gamma$-reflected Brownian motions with drift. The process for each component $i$ is given by $\widehat{X}\_i(t) = X\_i(t) - \gamma\_i \inf\_{s \le t} X\_i(s)$, where $X\_i(t) = B\_i(t) - c\_i t$ is a standard Brownian motion with drift, and $\gamma\_i \in (0, 1]$ is a reflection parameter. We derive a precise asymptotic formula for $\psi\_\gamma(u) = \mathbb{P}\{\exists\, t \geq 0, \forall\, i = 1,...,n \colon \widehat{X}\_i(t) > ub\_i\}$ as the threshold $u$ tends to infinity of the form $\psi\_\gamma(u) \sim C u^\zeta e^{-g^\* u^2 / 2}$, where $g^\*$ and $\zeta$ are given explicitly and $C \in (0, \infty)$ is a Pickands-type constant. This problem is motivated by applications in actuarial science, where $\psi(u)$ represents the simultaneous ruin of multiple insurance portfolios under a loss-carry-forward taxation scheme, and in queueing theory, where it corresponds to the simultaneous overflow of $n$ independent buffers.
Keywords
BibTeX
@article{Ievlev16112025,
author = {Pavel Ievlev and Svyatoslav Novikov},
title = {Extremes of uncorrelated gamma-reflected Brownian motions with trend},
journal = {Scandinavian Actuarial Journal},
volume = {0},
number = {0},
pages = {1--27},
year = {2025},
publisher = {Taylor \& Francis},
doi = {10.1080/03461238.2025.2587662}
}