# Probability of a stochastic process crossing a boundary in time interval

Suppose that we have a stochastic process $$X(t)$$: $$X(t) = \frac{1}{t}\int_{0}^{t} W(\tau) d\tau$$ where $$W(\tau)$$ is a Wiener process. What is the probability of $$X(t)$$ crossing a barrier $$\alpha$$ ($$\alpha > 0$$, also $$\alpha$$ is said to be large) at least once in time interval $$[t_1, t_2]$$?

• If you've found PFD of the original process you can find PDF of a stopped process using method of images, by just flipping it upside down with an offset. – Radost Apr 5 at 12:36