I have a question about the uniqueness of Doob-Meyer decomposition. For the sake of completeness, consider the following form of such a result:
Let ${\displaystyle Z}$ be a cadlag submartingale of class D. Then there exists a unique, increasing, predictable process ${\displaystyle A}$ with ${\displaystyle A_{0}=0}$ such that ${\displaystyle M_{t}=Z_{t}+A_{t}}$ is a uniformly integrable martingale.
I know that requiring predictability of the FV process $A$ in the decomposition yields a unique decomposition. But, what I do not know is: do we ever really need this uniqueness somewhere? For example: in results or in other constructions (of proofs?). Any example is welcome!