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Recall that F j = σ ( ε$_{j}$$^{, ε$_{j}$}$− $_{1}$, . . . , η$_{j}$ $^{, η$_{j}$}$− $_{1}$, . . . ). Define 

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for all x ∈ [0 , 1]. Then, X$_{n}$ ( x, t ) is adapted to F $_{n}$( x, t ). Moreover, the corresponding filtration is right-continuous. 

Let T$_{n}$ denote a canonical 2-stopping time for X$_{n}$ ( x, t ). In particular, we require { T$_{n}$ ≤ t } to be measurable with respect to F $_{n}$(1 , t ) = F ⌊ nt ⌋ . Define τ$_{n}$ := ⌊ nT$_{n}$ ⌋ . Note that 

with 

For 0 ≤ x ≤ 1 define 

In order to show (S1.2), we have to prove that 

Prior to the proof, we establish the following result: 

Lemma S1.2. We have E [ ˜ M$_{n}$ ( x ) ] = 0 for all x ∈ R . For x < y 2 

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