Portfolios under change of numéraire). Consider two assets with prices $S(t)$ and $N(t)$ given by
$$
\begin{aligned}
& S(t)=S(0) \exp \left\{\sigma \widetilde{W}(t)+\left(r-\frac{1}{2} \sigma^2\right) t\right\}, \\
& N(t)=N(0) \exp \left\{\nu \widetilde{W}(t)+\left(r-\frac{1}{2} \nu^2\right) t\right\},
\end{aligned}
$$
where $\widetilde{W}(t)$ is a one-dimensional Brownian motion under the risk-neutral measure $\widetilde{\mathbb{P}}$ and the volatilities $\sigma>0$ and $\nu>0$ are constant, as is the interest rate $r$. We define a third asset, the money market account, whose price per share at time $t$ is $M(t)=e^{r t}$.
Let us now denominate prices in terms of the numéraire $N$, so that the redenominated first asset price is
$$
\widehat{S}(t)=\frac{S(t)}{N(t)}
$$
and the redenominated money market account price is
$$
\widehat{M}(t)=\frac{M(t)}{N(t)} .
$$
According to Theorem 9.2.2, $d \widehat{S}(t)=(\sigma-\nu) \widehat{S}(t) d \widehat{W}(t)$, where $\widehat{W}(t)=\widetilde{W}(t)-$ $\nu t$.
(i) Compute the differential of $\frac{1}{N(t)}$.
(ii) Compute the differential of $\widehat{M}(t)$, expressing it in terms of $d \widehat{W}(t)$.
Consider a portfolio that at each time $t$ holds $\Delta(t)$ shares of the first asset and finances this by investing in or borrowing from the money market. According to the usual formula, the differential of the value $X(t)$ of this portfolio is
$$
d X(t)=\Delta(t) d S(t)+r(X(t)-\Delta(t) S(t)) d t .
$$
We define
$$
X(t)-\Delta(t) S(t)
$$