Let (M_{t}=int_{0}^{t}left(X_{s} d Y_{s}-Y_{s} d X_{s} ight)) where (X) and (Y) are two real-valued independent Brownian motions.

Let \(M_{t}=\int_{0}^{t}\left(X_{s} d Y_{s}-Y_{s} d X_{s}\right)\) where \(X\) and \(Y\) are two real-valued independent Brownian motions. Prove that

\[M_{t}=\int_{0}^{t} \sqrt{X_{s}^{2}+Y_{s}^{2}} d B_{s}\]

where \(B\) is a BM. Prove that

\[\begin{aligned}
X_{t}^{2}+Y_{t}^{2} & =2 \int_{0}^{t}\left(X_{u} d Y_{u}+Y_{u} d X_{u}\right)+2 t \\
& =2 \int_{0}^{t} \sqrt{X_{u}^{2}+Y_{u}^{2}} d \beta_{u}+2 t
\end{aligned}\]

where \(\beta\) is a Brownian motion, with \(d\langle B, \beta\rangle_{t}=0\).