Ruin Probability Approximation for Bidimensional Brownian Risk Model with Tax

Let $\mathbf{B}(t)=(B_1(t),B_2(t))$, $t\ge 0$ be a two-dimensional Brownian motion with independent components and define the $\mathbf{\gamma }$-reflected process $$\mathbf{X}(t)=(X_1(t),X_2(t))=(B_1(t)-c_{1}t-{\gamma }_1\underset{s_1\in [0,t]}{\inf }(B_1(s_1)-c_1{s}_1),B_2(t)-c_{2}t-{\gamma }_2\underset{s_2\in [0,t]}{\inf }(B_2(s_2)-c_2{s}_2)),$$ with given finite constants $c_1,c_2$ and ${\gamma }_1,{\gamma }_2\in [0,2)$. The goal of this paper is to derive the asymptotics of the ruin probability $$\mathbb{P}\{{\mathrm{\exists }}_{t\in [0,T]}:X_1(t)>u,X_2(t)>au\}$$ as $u\to \mathrm{\infty }$ and $T>0$.