Logarithmic asymptotics for probability of component-wise ruin in two-dimensional Brownian model

Krzysztof Debicki, Lanpeng Ji, Tomasz Rolski

Published 2019-06-21Version 1

Let ${\bf X}({\bf t})=(X_1(t),X_2(s)), {\bf t}=(t,s)$ be a correlated two-dimensional Brownian motion and let $\mu_1,\mu_2>0$ be two constants. In this contribution, we derive the logarithmic asymptotics \[ \log P\Bigl( \sup_{t\ge 0} \Bigl( X_1(t) - \mu_1 t\Bigr)> u, \ \sup_{s\ge 0} \Bigl( X_2(s) - \mu_2 s\Bigr)> u \Bigr),\qquad u\to\infty. \]