Let (Wt)(W_t)(Wt) be a standard Brownian motion and let Mt=max0≤u≤tWuM_t = \max_{0 \le u \le t} W_uMt=max0≤u≤tWu be its running maximum. Find the density fMt(y)f_{M_t}(y)fMt(y) of MtM_tMt for y≥0y \ge 0y≥0.
Here Φ\PhiΦ denotes the standard normal cumulative distribution function and φ(z)=12πe−z2/2\varphi(z) = \frac{1}{\sqrt{2\pi}}e^{-z^2/2}φ(z)=2π its density.