Let W W be a Brownian motion, F F its natural filtration and M t = sup

Let $W$ be a Brownian motion, $F$ its natural filtration and $M_t={sup}_{s\le t}{W}_s$. Prove that, for $t<1$,

$$E(f\left(M_1\right)\mid F_t)=F(1-t,W_t,M_t)$$

with

$F(s,a,b)=\sqrt{\frac{2}{\pi s}}\left(f\right(b){\int }_0^{b-a}{e}^{-u^2/\left(2s\right)}du+{\int }_b^{\infty }f(u\left)\exp (-\frac{(u-a{)}^2}{2s})du\right)$.

Note that

$$\underset{s\le 1}{sup}{W}_s=\underset{s\le t}{sup}{W}_s\vee \underset{t\le s\le 1}{sup}{W}_s=\underset{s\le t}{sup}{W}_s\vee ({\hat{M}}_{1-t}+W_t)$$

where ${\hat{M}}_s={sup}_{u<s}{\hat{W}}_u$ for ${\hat{W}}_u=W_{u+t}-W_t$.