Ruin probability approximation for bidimensional Brownian risk model with tax

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