a) Use the (two-dimensional, deterministic) chain rule \(d(F \circ G)=F^{\prime} \circ G d G\) to deduce the formula for integration by parts for Stieltjes integrals:

\[\int_{0}^{t} f(s) d g(s)=f(t) g(t)-f(0) g(0)-\int_{0}^{t} g(s) d f(s) \quad \text { for all }f, g \in \mathcal{C}^{1}([0, \infty), \mathbb{R})\]

b) Use the Itô formula for a Brownian motion \(\left(b_{t}, \beta_{t}\right)_{t \geqslant 0}\) in \(\mathbb{R}^{2}\) to show that

\[\int_{0}^{t} b_{s} d \beta_{s}=b_{t} \beta_{t}-\int_{0}^{t} \beta_{s} d b_{s}, \quad t \geqslant 0\]

What happens if \(b\) and \(\beta\) are not independent?