There are two random walks,
$S^t_i=S^{t-1}_i+ X_i^t$ for $i=1,2$, $X^t_i$ i.i.d
they have boundaries $h_1$ and $h_2$ respectively. I'm wondering if it's possible to calculate the probability that one hits its boundary before the other. i.e.
$\sum _{t=0}^\infty \mathrm{Pr}(S_t^1\geq h_1|S_\tau^2< h_2 $ for any $ \tau \leq t)$
or
$\mathrm{Pr}(\min (t: {S_t^1\geq h_1})<\min (t: {S_t^2\geq h_2}))$.
This problem seems to be bery practical, it's like determining the winning probability of two runners with different speed and destination. Any reference is appreciated! Thank you!