= Oppositely driven Bessel exit probability
{title2=$\mathbb P(\zeta_X<\zeta_Y)=I_{y/(x+y)}(1-2a,1-2a)$}
For $0<a<1/2$, couple $dX=dB+aX^{-1}dt$ and $dY=-dB+aY^{-1}dt$ from positive $x,y$, killing each at zero. Their sum has no Brownian term and increases, so simultaneous extinction is impossible. The ratio $U=Y/(X+Y)$ has time-changed generator $\frac12\partial_u^2+a(1-2u)[u(1-u)]^{-1}\partial_u$. Its scale density is $u^{-2a}(1-u)^{-2a}$, integrable at both endpoints. Optional stopping of its normalized scale function at the first extinction gives the displayed regularized <incomplete beta function>. At $a=1/3$ this is <Cardy's formula>.
Back to article page
