(Correlation under change of measure). Consider the multidimensional market model of Subsection 5.4.2, and let $B_i(t)$ be defined by (5.4.7). Assume that the market price of risk equations (5.4.18) have a solution $\theta_1(t), \ldots, \Theta_d(t)$, and let $\widetilde{\mathbb{P}}$ be the corresponding risk-neutral measure under which
$$
\widetilde{W}_j(t)=W_j(t)+\int_0^t \Theta_j(u) d u, \quad j=1, \ldots, d,
$$
are independent Brownian motions.
(i) For $i=1, \ldots, d$, define $\gamma_i(t)=\sum_{j=1}^d \frac{\sigma_{0,}(t) \theta_j(t)}{\sigma_i(t)}$. Show that
$$
\widetilde{B}_i(t)=B_i(t)+\int_0^t \gamma_i(u) d u
$$
is a Brownian motion under $\widetilde{\mathbb{P}}$.
(ii) We saw in (5.4.8) that
$$
d S_i(t)=\alpha_i(t) S_i(t) d t+\sigma_i(t) S_i(t) d B_i(t), \quad i=1, \ldots, m .
$$
Show that
$$
d S_i(t)=R(t) S_i(t) d t+\sigma_i S_i(t) d \tilde{B}_i(t), \quad i=1, \ldots, m
$$