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Suppose $W^{\mu}_t$ is a Brownian motion with drift $\mu$ and $Y_t$ represents the reflecting process of $x+W^{\mu}_t$ ($0<x<b$) which is reflected  at $b$. 

Suppose $0<x<b$. What is the joint distribution/density for $Y_t,\max_{0\le s\le t}Y_s$? Or what is $P(Y_t\in dy,\max_{0\le s\le t}Y_t>0)$?

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