Let \(X_{t}=M_{t}+A_{t}\) and \(X_{t}=M_{t}^{\prime}+A_{t}^{\prime}\) where \(M, M^{\prime}\) are continuous local martingales and \(A, A^{\prime}\) are continuous processes which have locally a.s. bounded variation. Show that \(M-M^{\prime}\) and \(A-A^{\prime}\) are a.s. constant.

Have a look at Corollary 17.3.

Data From Corollary 17.3

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17.3 Corollary. Let (Xt, F)o be an L2 martingale such that both (X - At, Ft)120 and (X-A, F)20 are martingales for some adapted processes (A+)20 and (A)120 having continuous paths of bounded variation on compact sets and Ap = A = 0. Then A and A' are indistinguishable: P(A = A vt > 0) = 1. Proof. By assumption A: - A = (x-A)-(x- Ar), i.e. A - A' is a martingale with continuous paths which are of bounded variation on compact sets. The claim follows from Proposition 17.2. The next step is to show that Theorem 17.1 holds for uniformly bounded martingales M M. This requires two lemmas. Let II = (0 to