Prove the result recalled in Comment 1.4.1.6.

If \(W_{T}^{(i)}\) could be written as \(\int_{0}^{T} \phi_{s}^{(i)} d B_{s}\) for \(i=1,2\), the properties of \(\phi^{(i)}\) would lead to a contradiction.

Comment 1.4.1.6:

A Brownian filtration is large enough to contain a strictly smaller Brownian filtration. On the other hand, if the processes \(W^{(i)}, i=1,2\) are independent real-valued Brownian motions, it is not possible to find a real-valued Brownian motion \(B\) such that \(\sigma\left(B_{s}, s \leq t\right)=\sigma\left(W_{s}^{(1)}, W_{s}^{(2)}, s \leq t\right)\). This will be proved using the predictable representation theorem.