Doob Decomposition Theorem in Williams is working backward? Unique modulo indistinguishability?

Question

Probability with Martingales

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This is my understanding of what is going on in the proof above:

1. We first assume $X$ has such Doob Decomposition in order to figure out what $A$ to use in order to prove the Doob Decomposition.

2. The details of the proof are left to the reader namely:

3. The proof that $A$ is previsible (obvious).

4. The proof that $M = X - X_0 - A$ is a martingale null at 0.

5. The details of unique modulo indistinguishability

Is that right?

How can we prove #5?

Answer 1

For 5.: The key is this: A previsible martingale is constant in time. In more detail, if $X$ admits a second decomposition $X_n=X_0+M'_n+A'_n$ then by subtracting you find that $M_n-M'_n$($=A'_n-A_n$) is both a martingale and previsible, hence constant in time a.s., whence $M_n-M'_n=M_0-M'_0=0-0=0$ for all $n$, a.s.