[X, Y]. A consequence of this is that the quadratic variation process of a stochastic integral is equal to an integral of a quadratic variation process, Integration by parts As with ordinary calculus, integration by parts is an important result in stochastic calculus. The integration by parts formula for the Itô integral differs from the standard result due to the inclusion of a quadratic covariation term. This term comes from the fact that Itô calculus deals with processes with non-zero quadratic variation, which only occurs for infinite variation processes (such as Brownian motion). If X and Y are semimartingales then where [X, Y] is the quadratic covariation process. The result is similar to the integration by parts theorem for the Riemann–Stieltjes integral but has an additional quadratic variation term. Itô's lemma Itô's lemma is the version of the chain rule or change of variables formula which applies to the Itô integral. It is one of the most powerful and frequently used theorems in stochastic calculus. For a continuous n-dimensional semimartingale X = (X1,...,Xn) and twice continuously differentiable function f from Rn to R, it states that f(X) is a semimartingale and, This differs from the chain rule used in standard calculus due to the term involving the quadratic covariation [Xi,Xj ]. The formula can be generalized to include an explicit time-dependence in and in other ways (see Itô's lemma). Martingale integrators Local martingales An important property of the Itô integral is that it preserves the local martingale property. If M is a local martingale and H is a locally bounded predictable process then H · M is also a local martingale. For integrands which are not locally bounded, there are examples where H · M is not a local martingale. However, this can only occur when M is not continuous. If M is a continuous local martingale then a predictable process H is M-integrable if and only if for each t, and H · M is always a local martingale. The most general statement for a discontinuous local martingale M is that if (H2 · [M])1/2 is locally integrable then H · M exists and is a local martingale. Square integrable martingales For bounded integrands, the Itô stochastic integral preserves the space of square integrable martingales, which is the set of càdlàg martingales M such that E[Mt2] is finite for all t. For any such square integrable martingale M, the