Quadratic variation and martingale brackets

Assume throughout this lesson a real martingale starting at zero whose value is square integrable at every finite time. Define its square bracket as the sum of the squares of its discrete increments. It is adapted, increasing, and integrable at every finite time. Its time-k increment generally depends on the new observation at k and need not be known at k minus one. The word optional here emphasizes that it uses the realized increment rather than its previous-information average.

Expanding the change in the square of the martingale gives twice the previous value times the new increment, plus the squared increment. Subtracting the realized square-bracket increment removes the second term. The remaining product has conditional mean zero because the earlier value is known and the increment is conditionally centered. Cauchy–Schwarz makes the product integrable. Thus the martingale square minus its realized square bracket is an integrable martingale, although the square bracket itself usually is not predictable.

The angle bracket sums the conditional squared increments given the information immediately before each one. Every summand is nonnegative, integrable, and measurable before its step, so the process is predictable and increasing. Conditional centering of the difference between each realized square and its conditional expectation shows that square bracket minus angle bracket is a martingale. Combining this with the first square-compensation identity gives the martingale square minus angle bracket identity.

Equivalently the angle bracket is the predictable increasing component in the Doob decomposition of the squared martingale, normalized to start at zero. Uniqueness of that decomposition makes this compensator unique up to almost-sure equality at all integer times. At each fixed time the expected square equals the expected angle bracket, since the starting value was zero. Equal expectations do not imply equality on individual paths; the difference is itself a centered martingale whose randomness may be substantial.

For a fair sign walk, every squared increment is one, so both brackets equal deterministic time. More generally, a predictable amplitude multiplied by a fair sign has squared increment equal to the amplitude squared, already known before the step. The two brackets again agree, but can now both be random because the amplitude depends on the past. These examples should not be generalized to arbitrary increments whose conditional squared size fluctuates after the new observation.

The brackets summarize different aspects of data. The realized bracket can be computed from the observed martingale path; the predictable bracket requires a conditional model. Their difference can diagnose fluctuations around a conditional variance forecast, but agreement of expectations does not validate the forecast model by itself. In later limit theorems the predictable bracket describes the variance accumulated before each increment, while an additional small-jump condition prevents one exceptional increment from determining the entire terminal distribution.