Let (f) be a (bounded) function. Prove that [lim _{t ightarrow infty} sqrt{t} mathbb{E}left(fleft(M_{t} ight) mid

Let \(f\) be a (bounded) function. Prove that

\[\lim _{t \rightarrow \infty} \sqrt{t} \mathbb{E}\left(f\left(M_{t}\right) \mid \mathcal{F}_{s}\right)=c\left(f\left(M_{s}\right)\left(M_{s}-W_{s}\right)+F\left(M_{s}\right)\right)\]

where \(c\) is a constant and \(F(x)=\int_{x}^{\infty} d u f(u)\).

Write \(M_{t}=M_{s} \vee\left(W_{s}+\widehat{M}_{t-s}\right)\) where \(\widehat{M}\) is the supremum of a Brownian motion \(\widehat{W}\), independent of \(W_{u}, u \leq s\).