I am currently studying the Brownian motion and I am stuck with a problem related to the reflection principle.
What I am trying to calculate is the probability that a standard Brownian Motion $X_t$ returns to zero given that it starts in $X_{t_a} = a$ and ends in $X_{t_b} = b$. ($t_a,t_b,a,b > 0$)
That is:
$P [X_t = 0 \hspace{1 mm}for \hspace{1 mm}some \hspace{1 mm} t∈[t_a,t_b]|X_{t_a} = a, X_{t_b} = b], \quad t_a,t_b,a,b > 0.$
Any answer or comment is greatly appreciated, thanks!