While learning the proof of Burkholder-Gundy-Davis inequality, I had to review the statement and proof of Doob’s maximal inequality (one of my favorite results in elementary probability). In the proof of the latter, I had to recall the meaning of the stopped filtration , as a critical step involves the
where is the running max of , and is the first time exceeds .
The last inequality appears a bit mysterious to me (still). So I looked into optional sampling theorem on wiki. Unfortunately nothing about the stopped filtration is explained there, but I did find an interesting exercise that took me a bit of time:
Suppose S(N) is with probability one either 100 or 0 and that S(0) = 50. Suppose further there is at least a sixty percent probability that the price will at some point dip to below 40 and then subsequently rise to above 60 before time N. Prove that S(n) cannot be a martingale.
Under the excuse of giving my readers a relaxing time, here is how I approached the problem (perhaps the only way): consider the stopping time . Then first observe that almost surely, since has to reach either and or at and by continuity it has to go through or . However by the probability assumption, thus violating optional sampling theorem.
Back to the meaning of the stopped filtration: the definition I recall is iff . This definition is both opaque and natural. Then I thought maybe in the special case when the filtration is generated by a single process , can be alternatively defined as . This is the subject of a stackexchange post (see ref 3). The conclusion there seems that the alternative definition holds only when is cadlag (continue a droit limite a gauche), or in other words the filtration is progressively measurable. The proof doesn't seem so trivial. I am sure it's true in the discrete case though still need to find time to see the actual proof.