论文标题
二维布朗运动的分量极值的精确渐近学
Exact asymptotics of component-wise extrema of two-dimensional Brownian motion
论文作者
论文摘要
我们得出\ [p \ left（\ sup_ {t \ ge 0} \ bigl（x_1（t） - μ_1t \ bigr）> u，\ \ \ sup_ {s \ ge 0} \ bigl（x_2（x_2（x_2（s） - μ_2s \ bigr）， \]其中$（x_1（t），x_2（s））_ {t，s \ ge0} $是一种相关的二维布朗运动，相关性$ρ\ in [-1,1] $和$μ_1，$μ_1，μ_2> 0 $。看来$ρ$和$μ_1，μ_2$之间的播放导致了几种类型的渐近学。尽管渐近学中的指数作为$ρ$的函数是连续的，但人们可以根据$ρ$的范围观察到不同类型的预取子函数，这构成了相型过渡现象。
We derive the exact asymptotics of \[ P\left( \sup_{t\ge 0} \Bigl( X_1(t) - μ_1 t\Bigr)> u, \ \sup_{s\ge 0} \Bigl( X_2(s) - μ_2 s\Bigr)> u \right), \ \ u\to\infty, \] where $(X_1(t),X_2(s))_{t,s\ge0}$ is a correlated two-dimensional Brownian motion with correlation $ρ\in[-1,1]$ and $μ_1,μ_2>0$. It appears that the play between $ρ$ and $μ_1,μ_2$ 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.
