Simultaneous ruin probability for multivariate gaussian risk model

Let $\mathbf{\text{Z}}(t)=(Z_1(t),\dots ,Z_d(t){)}^{\mathrm{\top }},t\in \mathbb{R}$ where $Z_i(t),t\in \mathbb{R}$, $i=1,\mathrm{.}\mathrm{.}\mathrm{.},d$ are mutually independent centered Gaussian processes with continuous sample paths a.s. and stationary increments. For $\mathbf{\text{X}}(t)=A\mathbf{\text{Z}}(t),t\in \mathbb{R}$, where $A$ is a nonsingular $d\times d$ real-valued matrix, $$\mathbf{\text{u}}$, $\mathbf{\text{c}}$\in$\mathbb{R}$^d$and$T>0$ we derive tight bounds for $$\mathbb{P}\{{\mathrm{\exists }}_{t\in [0,T]}:{\cap }_{i=1}^d\{X_i(t)-c_{i}t>u_i\}\}$$ and find exact asymptotics as $(u_1,…,u_d)^{\top}= (u a_1,…, ua_d)^\top$and$u\to\infty$.