Uniqueness of an increasing square compensator ($M^2-A\text{ local martingale},A_0=0$)

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.