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I'm missing meaning of $w$ in typical SDE like $dX_t(w) = f_t(X_t(w)) + \sigma(X_t(w))dW_t$, in context of $w \in F_{xxx}$. Does it mean that both

  1. $w$ is one of events that could happen before moment $xxx$
  2. If we fix $w$ then this SDE becomes deterministic

?

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In general SDE's are defined on a probability space which consists of a triplet $(\Omega, P, B)$: the space $\Omega$, a probability measure $P$, and a sigma algebra $B$. In short, the sigma algebra consist on the set of all events that we can assign probability to.

For SDE's driven by Brownian Motion this probability space is the so called Wiener space, which consist of $\Omega = C([0,T])$, the space of continuous functions equipped with the uniform topology. The sigma algebra is the one generated by the open sets under this topology, and $P$ is the so called Weiner measure, which essentially says that the projection $$\pi( \omega ) = \omega(t), \quad \omega \in \Omega,$$ is a Brownian Motion. So for example, under this measure, $$ P \{ \omega \in \Omega: \omega \text{ is differentiable } \} = 0. $$

This is the $\omega$ sometimes you see written in the math books.

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