Multidimensional Brownian risk models with random trend

TL;DR

This paper derives asymptotic probabilities for multidimensional Brownian risk models with random trends, extending classical models by incorporating random vectors with bounded support and analyzing their extreme value behavior.

Contribution

It provides new asymptotic results for the probability of simultaneous exceedances in multidimensional Brownian risk models with random, bounded trend vectors.

Findings

Asymptotic formulas for exceedance probabilities as thresholds go to infinity.

Extension of classical Brownian risk models to include random trend vectors.

Conditions under which the asymptotics hold.

Abstract

Let \(\mathbf B(t)=(B_1(t), \dots,B_d(t))^\top\), \(t\in[0,T]\), \(d\geq 2\) be a \(d\)-dimensional Brownian motion with independent components and let \(\mathbf \eta=(\eta_1,\dots,\eta_d)^\top\) be a random vector independent of \(\mathbf B\) such that \[ \mathbb{P}{\mathbf K_{1}\leq\mathbf\eta\leq\vk K_{2}} =\mathbb{P}{K_{11}\leq\eta_1\leq K_{21},\dots,K_{1d}\leq\eta_d\leq K_{2d}}=1, \] where \(\mathbf K_1=(K_{11},\dots,K_{1d})^\top\) and \(\vk K_2=(K_{21},\dots,K_{2d})^\top\) are fixed \(d\)-dimensional vectors. The goal of this paper is to derive asymptotics of \[ \mathbb{P}{\exists_{t\in[0,T]}: X_1(t)>a_1u,\dots,X_d(t)>a_du}, \ \ \mathbf X(t)=\left(X_1(t),\dots,X_d(t)\right)^\top =A\mathbf B(t)-\mathbf\eta t \] as \(u\to\infty\) under certain restrictions on the random vector \(\mathbf\eta\) and constants \(a_1,\dots, a_d\).