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HiI've just finished am MSc in financial maths, thinking of doing a PhD in stochastic analysis or a related field. Had a few questions I can't find obvious answers to:1/ Given a filtration F_t, is it guaranteed that there exists a process X_t such that F_t is the natural filtration of X? Does X have any uniqueness properties?2/Is it possible to quantify the amount of information in a filtration at time t? (I was thinking maybe the supremum of the shannon entropies of all F_t measurable random variables...)3/If a local martingale has a dense countable subset of the real numbers as its fundemental sequence, is this sufficient to show that it's a martingale?I'll be back with more soon...

An easier one:Brownian motion is a uniformly integrable martingale, so it converges to a random variable as t -> infinity.Is there an explicit expression for W_infinity if W_t is brownian motion?

I very much doubt that BM is uniformly integrable ! For instance, E|B_t| -> infinity as t -> infinity, so it's not even bounded in L^1. And no, B_t does not converge to +- infinity, or to any other limit : it keeps oscillating between high and low values (it is a recurrent Markov process !).About your Q 1/ : I don't have the answer for a general process (X_t) but I know there has been some work done on characterising the complete filtrations that are generated by a continuous martingale (M_t), I'll try to find a reference (you can look in Revuz-Yor for starters).

yes, my reply was hasty -- I stand corrected.

Cheers moltabile, I'll take a look

QuoteOriginally posted by: moltabileI very much doubt that BM is uniformly integrable ! For instance, E|B_t| -> infinity as t -> infinity, so it's not even bounded in L^1Isn't E[B_t] = 0 for all t?

Yes, but that doesn't say a lot about E|B_t| of course. Imagine X_n taking value n with prob. 1/2 and -n with prob. 1/2, then E[X_n]=0 for all n while E|X_n| -> \infty as n \to \infty.Indeed Brownian motion is *not* uniformly integrable!

Yeah, I realised my previous post was a bit dumb shortly after I posted it!

1) If you don't put any conditions on the paths, I think it up to technicalities more or less boils down to the question of whether any sigma algebra can be generated by a random variable. Here's a junky counterexample, but I think it represents a fundamental obstruction that a given sigma algebra contains too much information. Take a sample space of cardinality at least aleph-1 (power set of reals), and let the sigma algebra be the power set, so the sigma algebra has cardinality aleph 2. If X is a RV, let X_a be the preimage of a real number a. Then the sigma algebra generated by X is contained in the the set generated by arbitrary unions/intersections of this disjoint partition of Omega. The cardinality of the {X_a} is the same as the cardinality of the reals, and it follows that the sigma algebra generated by X has cardinality at most aleph 1. It probably has the same cardinality as the reals though.3) If the sequence is increasing, it can't be dense in the reals, ie everything is bigger than the first number you choose.

Very interesting, thank you RDK