(Creating independent Brownian motions to represent correlated ones). Let $B_1(t), \ldots, B_m(t)$ be $m$ one-dimensional Brownian motions with
$$
d B_i(t) d B_k(t)=\rho_{i k}(t) d t \text { for all } i, k=1, \ldots, m,
$$
where $\rho_{i k}(t)$ are adapted processes taking values in $(-1,1)$ for $i \neq k$ and $\rho_{i k}(t)=1$ for $i=k$. Assume that the symmetric matrix
$$
C(t)=\left[\begin{array}{cccc}
\rho_{11}(t) & \rho_{12}(t) & \cdots & \rho_{1 m}(t) \\
\rho_{21}(t) & \rho_{22}(t) & \cdots & \rho_{2 m}(t) \\
\vdots & \vdots & & \vdots \\
\rho_{m 1}(t) & \rho_{m 2}(t) & \cdots & \rho_{m m}(t)
\end{array}\right]
$$
is positive definite for all $t$ almost surely. Because the matrix $C(t)$ is symmetric and positive definite, it has a matrix square root. In other words, there is a matrix
$$
A(t)=\left[\begin{array}{cccc}
a_{11}(t) & a_{12}(t) & \cdots & a_{1 m}(t) \\
a_{21}(t) & a_{22}(t) & \cdots & a_{2 m}(t) \\
\vdots & \vdots & & \vdots \\
a_{m 1}(t) & a_{m 2}(t) & \cdots & a_{m m}(t)
\end{array}\right]
$$
such that $C(t)=A(t) A^{\operatorname{tr}}(t)$, which when written componentwise is