# Questions about existence and uniqueness theorem for stochastic differential equations in Oksendal's SDE book

In Oksendal's SDE book, Theorem 5.2.1. (Existence and uniqueness theorem for stochastic differential equations) assumes $Z$ is a random variable which is independent of the sigma algebra $\mathcal F_\infty^{(m)}$ generated by $B_s, s \geq 0$ and such that $E |Z|^2 < \infty$. Why is there a need to have the solution $X_t$ to a SDE adapted to the filtration $\mathcal F^Z_t$ of a random variable $Z$ and the Brownian motion $B_s, s\leq t$, instead of the filtration of the Brownian motion $B_s, s\leq t$?

Does "(the SDE ... has) a t-continuous solution $X_t(\omega)$" means that the sample path of the process $X$ is continuous wrt $t$ a.s.?

Thanks and regards!

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