  We consider an obliquely reflected Brownian motion $Z$ with positive drift in
a quadrant stopped at time $T$, where $T:=\inf \{ t>0 : Z(t)=(0,0) \}$ is the
first hitting time of the origin. Such a process can be defined even in the
non-standard case where the reflection matrix is not completely-$\mathcal{S}$.
We show that in this case the process has two possible behaviors: either it
tends to infinity or it hits the corner (origin) in a finite time. Given an
arbitrary starting point $(u,v)$ in the quadrant, we consider the escape (resp.
absorption) probabilities $\mathbb{P}_{(u,v)}[T=\infty]$ (resp.
$\mathbb{P}_{(u,v)}[T<\infty]$). We establish the partial differential
equations and the oblique Neumann boundary conditions which characterize the
escape probability and provide a functional equation satisfied by the Laplace
transform of the escape probability. We then give asymptotics for the
absorption probability in the simpler case where the starting point in the
quadrant is $(u,0)$. We exhibit a remarkable geometric condition on the
parameters which characterizes the case where the absorption probability has a
product form and is exponential. We call this new criterion the dual skew
symmetry condition due to its natural connection with the skew symmetry
condition for the stationary distribution. We then obtain an explicit integral
expression for the Laplace transform of the escape probability. We conclude by
presenting exact asymptotics for the escape probability at the origin.
