/sci/ - Science & Math

>>8449662
Can someone PLEASE give me an explanation on filtrations and adapted processes?

From my understanding we take a stochastic process [math] X_t [/math] and then every sigma algebra [math] \mathcal{F}_t [/math] we just add the sets [math]\{ \omega \in \Omega: X_t(\omega) = X_t, \ \forall s < t \} [/math]. In words; we add a set with all the [math]\omega[/math]'s such that, from our observations, one of the [math]\omega[/math]'s in that set is the real path of the process, and of course we generate a new sigma algebra out of that set.

I'm looking for an intuitive explanation that isn't just "filtration is all the information known at time t".

>>8416217
I'm a mathematics undergraduate and I'm doing a bonus-project on stochastic processes. Right now I have gone over the Brownian motion (properties, Lévy construction, filtrations), and I at least want to go over the stochastic integral after this. Any recommendations for other things I can include? Or recommendations for literature?

Can someone explain to a retard what it means when a stochastic process is "adapted to a filtration".

I know intuitively that it means the process [math] \{ X_t \}_{t \geq s} [/math] is only dependent on the events that happened before time [math] s [/math], but the definition is some vague, measure theoretic definition and I really don't see how the definition ties in with the intuition.