Assume that (left(S_{t} ight)_{t in mathbb{R}_{+}})and (left(N_{t} ight)_{t in mathbb{R}_{+}})satisfy the stochastic differential equations [d S_{t}=r_{t} S_{t}

 Assume that \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\)and \(\left(N_{t}\right)_{t \in \mathbb{R}_{+}}\)satisfy the stochastic differential equations

\[d S_{t}=r_{t} S_{t} d t+\sigma_{t}^{S} S_{t} d W_{t}^{S}, \quad \text { and } \quad d N_{t}=\eta_{t} N_{t} d t+\sigma_{t}^{N} N_{t} d W_{t}^{N}\]

where \(\left(W_{t}^{S}\right)_{t \in \mathbb{R}_{+}}\)and \(\left(W_{t}^{N}\right)_{t \in \mathbb{R}_{+}}\)have the correlation

\[d W_{t}^{S} \cdot d W_{t}^{N}=ho d t\]

where \(ho \in[-1,1]\).

a) Show that \(\left(W_{t}^{N}\right)_{t \in \mathbb{R}_{+}}\)can be written as

\[W_{t}^{N}=ho W_{t}^{S}+\sqrt{1-ho^{2}} W_{t}, \quad t \geqslant 0\]

where \(\left(W_{t}\right)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion under \(\mathbb{P}^{*}\), independent of \(\left(W_{t}^{S}\right)_{t \in \mathbb{R}_{+}}\).

b) Letting \(X_{t}=S_{t} / N_{t}\), show that \(d X_{t}\) can be written as

\[d X_{t}=\left(r_{t}-\eta_{t}+\left(\sigma_{t}^{N}\right)^{2}-ho \sigma_{t}^{N} \sigma_{t}^{S}\right) X_{t} d t+\widehat{\sigma}_{t} X_{t} d W_{t}^{X}\]

where \(\left(W_{t}^{X}\right)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion under \(\mathbb{P}^{*}\) and \(\widehat{\sigma}_{t}\) is to be computed.