Assume that (left(B_{t}ight)_{t in mathbb{R}_{+}})and (left(W_{t}ight)_{t in mathbb{R}_{+}})are standard Brownian motions, correlated according to the It rule

Assume that \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)and \(\left(W_{t}ight)_{t \in \mathbb{R}_{+}}\)are standard Brownian motions, correlated according to the Itô rule \(d W_{t} \cdot d B_{t}=ho d t\) for \(ho \in[-1,1]\), and consider the solution \(\left(Y_{t}ight)_{t \in \mathbb{R}_{+}}\)of the stochastic differential equation \(d Y_{t}=\mu Y_{t} d t+\eta Y_{t} d W_{t}, t \geqslant 0\), where \(\mu, \eta \in \mathbb{R}\) are constants. Compute \(d f\left(S_{t}, Y_{t}ight)\), for \(f\) a \(\mathcal{C}^{2}\) function on \(\mathbb{R}^{2}\) using the bivariate Itô formula (4.26).