Hi,I have the following exercise where I am really stuck. Would really appreciate any help.Is the following statement true or false ?Let p>1, and X = (X_n)_{n integer >= 0} is a stochastic process in Lp, ie ||X||_p < oo, then there exist Y in Lp such that for all n>=0, E(Y|An) = Xn, where An is a filtration adapted to Xn.Thanks

A Doob martingale http://math.iisc.ernet.in/~manju/Browni ... .pdfQuoteA natural (and surprisingly useful!) way to construct martingales is to take an arbitrary random variable M ∈ L1(P) and any filtration Fn, and then define the sequence Mn = E[M|Fn] - this is called a "Doob martingale". The martingale convergence theorem is a converse of sorts to this statement, that any uniformly integrable martingale is a Doob martingale.// Regarding the "converse of sorts" part -- note that any UI martingale can be closed: http://books.google.com/books?id=mJkFuq ... closed"See also:http://www.cs.berkeley.edu/~sinclair/cs ... 321.pdfThe last one has a cool example:QuoteConsider a run of quicksort on a particular input: let Q be the number of comparisons. Let X1 be the first pivot, X2 the second, etc. Then Zi = E[Q | X1, ..., Xi] is a Doob martingale with respect to (Xi).

Hi PolterThanks for your response, from what I can see in your links, it looks like the doob martingale is basically a random martingale M that we take and then its projection on the filtration adapted to Mn is a martingale. But how can we build this M using the Mn ?Thanks

Ah, you need to obtain the closing random variable?Then, the converse part applies, take a look at Theorem 12 in Protter (which says that limit will work as the closing r.v.).// BTW, "M" is an ordinary, integrable random variable, not a "random martingale."

Unfortunately in his book he just enunciates theorem 12 but does not prove anything. Do you have any proof related to this or not ?

Corollary 2 under Theorem 34: http://ocw.mit.edu/courses/mathematics/ ... pdfRelying on:http://ocw.mit.edu/courses/mathematics/ ... ion_13.pdf // Defs. on pp. 51 & 52; see also Lemma 29http://ocw.mit.edu/courses/mathematics/18-175-theory-of-probability-fall-2008/lecture-notes/section_14.pdf // Th. 33

Thanks Polter for your great help !