Let \(W\) be a BM and \(\varphi\) be an adapted process.

(a) Prove that \(\int_{0}^{t} \varphi_{s} d W_{s}\) is a BM if and only if \(\left|\varphi_{s}\right|=1\), ds a.s.

(b) Assume now that \(\varphi\) is deterministic. Prove that \(W_{t}-\int_{0}^{t} d s \varphi_{s} W_{s}\) is a BM if and only if \(\varphi \equiv 0\) or \(\varphi \equiv \frac{1}{s}\), ds a.s..

The function \(\varphi\) satisfies, for \(t>s\),

\[\mathbb{E}\left(\left(W_{t}-\int_{0}^{t} d u \varphi_{u} W_{u}\right)\left(W_{s}-\int_{0}^{s} d u \varphi_{u} W_{u}\right)\right)=s\]

if and only if \(s \varphi_{s}=\varphi_{s} \int_{0}^{s} d u u \varphi_{u}\).