the predictable quadratic variation (angle bracket)
Given a square-integrable martingale M, the process M^2 is a submartingale (Jensen), so it has an upward trend; the predictable quadratic variation <M> is precisely that trend — the predictable increasing process that, subtracted from M^2, leaves a martingale. It is the angle-bracket object written <M>_t, and it measures the accumulated 'predictable energy' or expected squared fluctuation of M. It is the deterministic-looking clock that governs how fast a martingale spreads out, and it is the cornerstone quantity of the Ito isometry and the martingale central limit theorem.
Precisely: for a square-integrable cadlag martingale M with M_0 = 0, apply the Doob-Meyer decomposition to the submartingale M^2. This yields a unique predictable increasing process <M>, with <M>_0 = 0, such that M_t^2 - <M>_t is a martingale. Equivalently <M>_t is the compensator of M^2, and its increments capture the conditional expected squared increment: heuristically d<M>_t = E[(dM_t)^2 given F_{t-}]. The bilinear polarization <M, N> = (1/4)(<M+N> - <M-N>) is the predictable covariation, characterized by M_t N_t - <M, N>_t being a martingale. For continuous martingales the angle bracket <M> and the optional bracket [M] coincide; they differ only by the compensated jump part, [M]_t = <M>_t + (predictable-jump correction) in the presence of jumps.
Why it is central: <M>_t is the variance clock. For Brownian motion <B>_t = t exactly, which is the rigorous content of the (dB)^2 = dt rule and the reason the Ito integral satisfies the Ito isometry E[(integral H dM)^2] = E[integral H^2 d<M>]. Levy's characterization says a continuous martingale with <M>_t = t IS a Brownian motion. In the martingale CLT, normalized martingales whose <M> converges to a deterministic limit converge to a Gaussian with that variance. The honest distinction to keep straight: <M> is PREDICTABLE and is defined only when M^2 is integrable enough for Doob-Meyer (square-integrability, or after localization for local martingales); the closely related [M] is OPTIONAL, always exists for any semimartingale as a pathwise limit of sums of squared increments, and equals <M> only in the continuous case — confusing the two at jumps is a standard error.
For the compensated Poisson martingale M_t = N_t - lambda t, the predictable quadratic variation is <M>_t = lambda t (smooth, predictable, deterministic here), whereas the optional quadratic variation is [M]_t = N_t (it jumps by 1 at each arrival). They have the same expectation, E[M_t^2] = lambda t, but [M] is the actual squared-jump sum while <M> is its predictable compensator.
Angle bracket <M> = lambda t (predictable) versus square bracket [M] = N_t (optional): equal in mean, different as processes — they coincide only when M is continuous.
The angle bracket is defined via Doob-Meyer and so needs enough integrability (square-integrability, or localization); it equals the square bracket only for continuous martingales, and equating them in the presence of jumps is a frequent mistake.