The aim of this exercise is to compute, for \(t

Prove that

\[\mathbb{E}\left(h\left(W_{T}\right) \mathbb{1}_{\left\{T

where

\[\Phi(x)=\sqrt{\frac{2}{\pi}} \int_{0}^{x} \exp \left(-\frac{u^{2}}{2}\right) d u\]

Define \(k(w)=h(w) \Phi(|w| / \sqrt{1-T})\). Prove that \(\mathbb{E}\left(k\left(W_{T}\right) \mid \mathcal{F}_{t}\right)=\widetilde{k}\left(t, W_{t}\right)\), where

\[\begin{aligned}\widetilde{k}(t, a) & =\mathbb{E}\left(k\left(W_{T-t}+a\right)\right) \\
& =\frac{1}{\sqrt{2 \pi(T-t)}} \int_{\mathbb{R}} h(u) \Phi\left(\frac{|u|}{\sqrt{1-T}}\right) \exp \left(-\frac{(u-a)^{2}}{2(T-t)}\right) d u .\end{aligned}\]