Existence and uniqueness of SDE, is the independence requirement needed?

Question

In Bernt Øksendals Stochastic differential equations he has this theorem in chapter 5:

$\\\\\\$ However, in the proof I can not see where he uses the independence condition I marked in red. Do you know if this theorem also holds if we do not assume independence? The proof is long, so I might have missed where he uses it. So do you know if the theorem holds without this? If you are familiar with the proof, do you see where he uses this condition?

$\\\\\\$

reference: http://th.if.uj.edu.pl/~gudowska/dydaktyka/Oksendal.pdf

chapter 5.2

Answer 1

On the right side of the case $k=0$ of formula (5.1.12) you have the Ito integral $\int_0^t\sigma(s,Z)\,dB_s$. For this to make sense we need $B$ to be a Brownian motion with respect to the filtration $(\mathcal F^Z_t)$, and the simplest way to ensure this is to assume that $Z$ is independent of $B$.