Prove that, more generally than (5.2.8), the dual predictable projection of (int_{0}^{t} fleft(B_{s}^{(u)} ight) d s) is

Prove that, more generally than (5.2.8), the dual predictable projection of \(\int_{0}^{t} f\left(B_{s}^{(u)}\right) d s\) is \(\int_{0}^{t} \mathbb{E}\left(f\left(B_{s}^{(u)}\right) \mid \mathcal{G}_{s}^{(u)}\right) d s\) and that

\[\mathbb{E}\left(f\left(B_{s}^{(u)}\right) \mid \mathcal{G}_{s}^{(u)}\right)=\frac{f\left(B_{s}^{(u)}\right) e^{u B_{s}^{(u)}}+f\left(-B_{s}^{(u)}\right) e^{-u B_{s}^{(u)}}}{2 \cosh \left(u B_{s}^{(u)}\right)}\]

\(2 u \int_{0}^{t} B_{s}^{(u)} d s\) is \(2 \int_{0}^{t} d s \psi\left(u B_{s}^{(u)}\right)\). (5.2.8)