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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 $[t_1, t_2]$?

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  • $\begingroup$ 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. $\endgroup$ – Radost Apr 5 at 12:36

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