Quadratic Variation - Martingales

Martingales

All càdlàg martingales, and local martingales have well defined quadratic variation, which follows from the fact that such processes are examples of semimartingales. It can be shown that the quadratic variation of a general local martingale M is the unique right-continuous and increasing process starting at zero, with jumps Δ = ΔM2, and such that M2 − is a local martingale.

A useful result for square integrable martingales is the Itō isometry, which can be used to calculate the variance of Ito integrals,

This result holds whenever M is a càdlàg square integrable martingale and H is a bounded predictable process, and is often used in the construction of the Itō integral.

Another important result is the Burkholder–Davis–Gundy inequality. This gives bounds for the maximum of a martingale in terms of the quadratic variation. For a continuous local martingale M starting at zero, with maximum denoted by M_t* ≡sup_s≤t|M_s|, and any real number p > 0, the inequality is

Here, c_p < C_p are constants depending on the choice of p, but not depending on the martingale M or time t used. If M is a continuous local martingale, then the Burkholder–Davis–Gundy inequality holds for any positive value of p.

An alternative process, the predictable quadratic variation is sometimes used for locally square integrable martingales. This is written as <M>_t, and is defined to be the unique right-continuous and increasing predictable process starting at zero such that M2 − <M> is a local martingale. Its existence follows from the Doob–Meyer decomposition theorem and, for continuous local martingales, it is the same as the quadratic variation.