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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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Hey in regards to your first question the filtration is generated by Z and B(s,w) for s<=t. In this case Z is the initial value and it is assumed to be a RV independent of the brownian motion.

In regards to your second questions saying it has a t-continuous solution means there exists a solution to the SDE that has continuous paths. Remember a DE (whether deterministic or stochastic) may have many solutions. However, if the conditions for the theorem are met their exists a unique t-continuous solution, ie a solution X(t,w) where for any fixed w (except on a set of measure 0) the sample paths are continuous w.r.t time. Note: I believe the second condition is sufficient for uniqueness and the first is sufficient for existence, but this should be verified.

Hope this helped!

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