 Predictable quadratic variation

For a zero-starting square-integrable martingale, the predictable quadratic variation is the predictable increasing process $A$ for which $M^2-A$ is a martingale. For a compensated jump process it integrates squared jump sizes against their rates. Unlike optional quadratic variation, it need not equal the sum of squared realized jumps. For continuous local martingales, the two brackets coincide after localization.

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