I have the following problem thats been mind boggling me for some time. If i can get some help, i would greatly appreciate it.

Suppose $T_A$ is a stopping time of a process from state A to state 0 and suppose $T_B$ is a stopping time of the process from state B to state 0. I need to find $P(T_A < T_B)$.

I read some articles online that mentioned that if $T = T_A - T_B$, then $G_T(t) = P(T <=t)$ is what i need. The paper suggested taking inverse laplace of a CDF to obtain the CDF and evaluate it at 0. It first suggested finding laplace transform of T, which is given by ${T} = E(e^{-ST}) = L[T_A](s) L[T_B](-s)$

Then it suggested taking laplace of $G_T(t)$ i.e $L[G_T(t)]$

However, $L{G_T(t)} = L[T] / s = (L[T_A](s) L[T_B](-s))/s$

Then the paper suggests taking the inverse of above evaluated at 0 to get $P(T_A < T_B)$.

Question, when i use a numerical procedure to obtain inverse laplace, wouldn't putting 0 in place of t give me 0 as this is a CDF and this is exactly what i am getting.

Thanks