= Uniqueness of an increasing square compensator
{title2=$M^2-A\text{ local martingale},\quad A_0=0$}
For a <continuous local martingale> $M$, at most one continuous adapted increasing process $A$ starting at zero makes $M^2-A$ a <local martingale>. The difference of two candidates is a continuous <finite-variation process> and a <local martingale>, hence constant; its initial value is zero. Without the initial-value normalization only uniqueness up to an initial constant is possible.
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