quadratic variation process [M] is integrable, and the Itô isometry states that This equality holds more generally for any martingale M such that H2 · [M]t is integrable. The Itô isometry is often used as an important step in the construction of the stochastic integral, by defining H · M to be the unique extension of this isometry from a certain class of simple integrands to all bounded and predictable processes. p-Integrable martingales For any p > 1, and bounded predictable integrand, the stochastic integral preserves the space of p-integrable martingales. These are càdlàg martingales such that E(|Mt|p) is finite for all t. However, this is not always true in the case where p = 1. There are examples of integrals of bounded predictable processes with respect to martingales which are not themselves martingales. The maximum process of a càdlàg process M is written as M*t = sups ≤t |Ms|. For any p ≥ 1 and bounded predictable integrand, the stochastic integral preserves the space of càdlàg martingales M such that E[(M*t)p] is finite for all t. If p > 1 then this is the same as the space of p-integrable martingales, by Doob's inequalities. The Burkholder–Davis–Gundy inequalities state that, for any given p ≥ 1, there exist positive constants c, C that depend on p, but not M or on t such that for all càdlàg local martingales M. These are used to show that if (M*t)p is integrable and H is a bounded predictable process then and, consequently, H · M is a p-integrable martingale. More generally, this statement is true whenever (H2 · [M])p/2 is integrable. Existence of the integral Proofs that the Itô integral is well defined typically proceed by first looking at very simple integrands, such as piecewise constant, left continuous and adapted processes where the integral can be written explicitly. Such simple predictable processes are linear combinations of terms of the form Ht = A1{t > T} for stopping times T and FT-measurable random variables A, for which the integral is This is extended to all simple predictable processes by the linearity of H · X in H. For a Brownian motion B, the property that it has independent increments with zero mean and variance Var(Bt) = t can be used to prove the Itô isometry for simple predictable integrands, By a continuous linear extension, the integral extends uniquely to all predictable integrands satisfying