Multidimensional Brownian risk models with random trend (2407.15995v1)

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).