This is what I thought initially, just a traditional enlargement of the space and filtration so as to to have a Brownian motion to work with. What is making me think this notion of “adjunction” is different is the context in which the term appears in the theorem. That is, as in the theorem, we only need to adjunct a Brownian motion if the measure (P x Lebesgue) of {(omega, t) : psi_t (omega) = 0} is zero where (psi_t) is the nondecreasing process that defines the 2nd moment of the increments of (X_t) — it’s not clear to me that this has any connection to the underlying filtered probability space not being able to support a Brownian motion, but I don’t know for certain. I will check out Doob’s text, thanks!
Hello Qui-Gon, I got a chance to look at the specific theorem statement the author quote's from Doob's book and I am confident it is the same definition you and I are thinking of. I'm sure you can construct a space if given the martingale conditions and a process psi_t which has adapted paths, is measurable w.r.t dtdP measure, and is non-negative (however, not necessarily vanishing almost nowhere) such that no Brownian Motion can exist on the space and thus the hypothesis can not hold. Consider a space in which there can't exist a process that is vanishing almost nowhere. Now, if there can exist a B.M. (adjunction comes into play) in the space and the other conditions hold, then the hypothesis holds in a similar fashion to the original proof that considers psi_t to vanish almost nowhere. Again, feel free to correct me.
