Proof of Theorem 1.1 Let s > 0 and Γ s ∈ F s fixed. Using the definition of Q F x,t (Γ s ) and the Markov property we have : Property (1.7) and inequality (1.8) allow us to apply the dominated convergence theorem : 1.3 In this paper, we investigate four cases of examples involving respectively for (A t ) : • the unilateral maximum (resp. minimum) S t (resp. I t ) : We also consider, in the same case study the two-dimensional process (S t , t). • (L 0 t ; t ≥ 0) the local time at 0 of (X t ) t≥0

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... . • The triplet ((S t , I t , L 0 t ); t ≥ 0). • (D t ; t ≥ 0) the number of down-crossings of X from level b to level a. We observe that, in all cases, the function M s (y 0 , f ; y) may be written as : for some function M and some α ∈ R; in fact α = 0, except for the case 1, b), as shown below. Since s → M s (y 0 , f ; Y s ) is a P x0 -martingale, it is clear that s → M (f ; Y s )e αs is also a P x0 -martingale. The results are summarized in the following Table :