I am dealing with a question listed below. I am trying to use the running maximum of Brownian motion to deal with the problem, but it does not work out.

Let $ \tau_{M}=\inf\{t;W(t)=M\},M>0,$ and

$X(t)=\begin{cases} W(t) & \text{if }t<\tau_{m}\\ M & \text{if }t\geq\tau_{m} \end{cases}$

Find the density function of $X(t)$.


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