(Creating correlated Brownian motions from independent ones). Let $\left(W_1(t), \ldots, W_d(t)\right)$ be a $d$-dimensional Brownian motion. In particular, these Brownian motions are independent of one another. Let $\left(\sigma_{i j}(t)\right)_{i=1, \ldots, m ; j=1, \ldots, d}$ be an $m \times d$ matrix-valued process adapted to the filtration associated with the $d$-dimensional Brownian motion. For $i=1, \ldots, m$, define
$$
\sigma_i(t)=\left[\sum_{j=1}^d \sigma_{i j}^2(t)\right]^{\frac{1}{2}},
$$
and assume this is never zero. Define also
$$
B_i(t)=\sum_{j=1}^d \int_0^t \frac{\sigma_{i j}(u)}{\sigma_i(u)} d W_j(u) .
$$
(i) Show that, for each $i, B_i$ is a Brownian motion.
200
4 Stochastic Calculus
(ii) Show that $d B_i(t) d B_k(t)=\rho_{i k}(t)$, where
$$
\rho_{i k}(t)=\frac{1}{\sigma_i(t) \sigma_k(t)} \sum_{j=1}^d \sigma_{i j}(t) \sigma_{k j}(t) .
$$