# Stochastic processes – Probability of calculating Brownian motion on the boundary

Let $$theta = min {t mid X_ {t} + t not in (0,1) }$$ represent the first time when $$Y_ {t} = X_ {t} + t$$ does not belong to $$(0, 1)$$. Calculate $$P_ {x} {Y _ { theta} = 0 }$$, or $$X_ {0} = x in (0, 1)$$

I'm trying to calculate the probability that we would reach the $$0$$ limit before the $$1$$ limit, I believe. I could not find too many similar examples and would appreciate help. I do not really know how to start.

Some examples I saw only deal with a Brownian motion striking the limit, but there is a $$t$$ term here. I would appreciate a lot of help because I study for an exam. I think this problem is related to Ito's lemma.