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4 is devoted to the characterization of all such martingales; in particular, we show that Xγ = 0 if and only if E [X|Fγ ] = 0. 6, we investigate how, in general, the quantities Xγ and E [X|Fγ ] differ. 1 Some Quantities Associated with γ Most of the results in this section may be read from Tables 1α and 2α; however, the following discussion is essentially self-contained. 1 This is different from, but related to, Williams’ path decomposition [Wil74] before and after γT1 = sup {t < T1 ; Bt = 0} R. Mansuy and M.

1 If (C) is not satisfied, we use the optional compensator, instead of the predictable one. 2 D. Williams’ Example of a Pseudo-stopping Time D. Williams [Wil02] provided an example of a non-stopping time ρ such that for every bounded martingale (Mt ; t ≥ 0), E [M∞ ] = E [Mρ ] Such a time will be called here a pseudo-stopping time. Before characterizing these times, we detail D. Williams’ original example: ρ = sup{t ≤ γT1 , Bt = St } where T1 is the first hitting time of 1 by B and γs still denotes the last zero of B before s.

A number of results about the Brownian meander are presented in [BY88]. 2 Some Examples of Martingales which Vanish on Z = {t; Bt = 0} This section aims at describing5 M0 the set of local martingales (Xt ; t ≥ 0) which vanish on the zero set of a given Brownian motion (Bt ; t ≥ 0). A number of examples are easily discovered. 2 Let us look for elements of M0 of the form6 (Bt Ht ; t ≥ 0) where H is a semimartingale with canonical decomposition N + V . Then, o’s formula: (Bt Ht ; t ≥ 0) is a local martingale if and only if, using Itˆ 4 5 6 A Rayleigh distributed random variable can be obtained from two independent √ (law) √ 2e.

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