We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Let 
${\mathrm{d}} X(t) = -Y(t) \, {\mathrm{d}} t$, where Y(t) is a one-dimensional diffusion process, and let 
$\tau(x,y)$ be the first time the process (X(t), Y(t)), starting from (x, y), leaves a subset of the first quadrant. The problem of computing the probability 
$p(x,y)\,:\!=\, \mathbb{P}[X(\tau(x,y))=0]$ is considered. The Laplace transform of the function p(x, y) is obtained in important particular cases, and it is shown that the transform can at least be inverted numerically. Explicit expressions for the Laplace transform of 
$\mathbb{E}[\tau(x,y)]$ and of the moment-generating function of 
$\tau(x,y)$ can also be derived.
