Ruin Probability Approximation for Bidimensional Brownian Risk Model with Tax

Let B(t)=(B₁(t), B₂(t)), t≥ 0 be a two-dimensional Brownian motion with independent components and define the γ-reflected process X(t)=(X₁(t),X₂(t))=(B₁(t)-c₁t-γ₁infs₁∈[0,t](B₁(s₁)-c₁s₁),B₂(t)-c₂t-γ₂infs₂∈[0,t](B₂(s₂)-c₂s₂)), with given finite constants c₁,c₂,γ₁,γ₂ and γ₁,γ₂∈[0,1]. The goal of this paper is to derive the asymptotics of ruin probability P\∃t∈[0,T]: X₁(t)>u,X₂(t)>au\ as u→∞ taking a≤ 1 and $T>0$.