continuous linear extension can be used to construct the integral for all left-continuous and adapted integrands with right limits everywhere (caglad or L-processes). This is general enough to be able to apply techniques such as Itô's lemma . Also, a Khintchine inequality can be used to prove the dominated convergence theorem and extend the integral to general predictable integrands . Differentiation in Itô calculus The Itô calculus is first and foremost defined as an integral calculus as outlined above. However, there are also different notions of "derivative" with respect to Brownian motion: Malliavin derivative Malliavin calculus provides a theory of differentiation for random variables defined over Wiener space, including an integration by parts formula . Martingale representation The following result allows to express martingales as Itô integrals: if M is a square-integrable martingale on a time interval [0, T] with respect to the filtration generated by a Brownian motion B, then there is a unique adapted square integrable process α on [0, T] such that almost surely, and for all t ∈ [0, T] . This representation theorem can be interpreted formally as saying that α is the "time derivative" of M with respect to Brownian motion B, since α is precisely the process that must be integrated up to time t to obtain Mt − M0, as in deterministic calculus. Itô calculus for physicists In physics, usually stochastic differential equations (SDEs), such as Langevin equations, are used, rather than stochastic integrals. Here an Itô stochastic differential equation (SDE) is often formulated via where is Gaussian white noise with and Einstein's summation convention is used. If is a function of the xk, then Itô's lemma has to be used: An Itô SDE as above also corresponds to a Stratonovich SDE which reads SDEs frequently occur in physics in Stratonovich form, as limits of stochastic differential equations driven by colored noise if the correlation time of the noise term approaches zero. For a recent treatment of different interpretations of stochastic differential equations see for example . Itô interpretation and supersymmetric theory of SDEs In the supersymmetric theory of SDEs, stochastic evolution is defined via stochastic evolution operator (SEO) acting on differential forms of the phase space. The Itô-Stratonovich dilemma takes the form of the ambiguity of the operator ordering that arises on the way from the path integral to the operator representation of stochastic evolution. The Itô interpretation corresponds to the