Let \(B_{t}=\left(b_{t}, \beta_{t}ight), t \geqslant 0\) be a \(\mathrm{BM}^{2}\) and set \(r_{t}:=\left|B_{t}ight|=\sqrt{b_{t}^{2}+\beta_{t}^{2}}\).

a) Show that the stochastic integrals \(\int_{0}^{t} b_{s} / r_{s} d b_{s}\) and \(\int_{0}^{t} \beta_{s} / r_{s} d \beta_{s}\) exist.

b) Show that \(W_{t}:=\int_{0}^{t} b_{s} / r_{s} d b_{s}+\int_{0}^{t} \beta_{s} / r_{s} d \beta_{s}\) is a \(\mathrm{BM}^{1}\).