We derive the exact asymptotics of

$ {\mathbb {P} \left \{ \underset {t\ge 0}{\sup } \left (X_{1}(t) - \mu _{1} t\right )> u, \ \underset {s\ge 0}{\sup } \left (X_{2}(s) - \mu _{2} s\right )> u \right \} },\ \ u\to \infty , $

where (X₁(t), X₂(s))_{t, s≥ 0} is a correlated two-dimensional Brownian motion with correlation ρ ∈ [− 1,1] and μ₁, μ₂ > 0. It appears that the play between ρ and μ₁, μ₂ leads to several types of asymptotics. Although the exponent in the asymptotics as a function of ρ is continuous, one can observe different types of prefactor functions depending on the range of ρ, which constitute a phase-type transition phenomena.

中文翻译：

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二维布朗运动的分量极值的精确渐近性

我们推导了的精确渐近性

$ {\ mathbb {P} \ left \ {\ underset {t \ ge 0} {\ sup} \ left（X_ {1}（t）-\ mu _ {1} t \ right）> u，\ \ underset {s \ ge 0} {\ sup} \ left（X_ {2}（s）-\ mu _ {2} s \ right）> u \ right \}}，\ \ u \ to \ infty，$

其中（X ₁（吨），X ₂（小号））_{吨，š ≥0}是具有相关性的相关二维布朗运动ρ＆Element; [ - 1,1]和μ ₁，μ ₂ > 0。看来，之间发挥ρ和μ ₁，μ _2根引线到几种类型的渐近。尽管渐近性的指数随ρ的变化是连续的，但根据ρ的范围，人们可以观察到不同类型的因子函数，构成了相型过渡现象。