predictable processes form the smallest class that is closed under taking limits of sequences and contains all adapted left-continuous processes. If H is any predictable process such that ∫0t H2 ds < ∞ for every t ≥ 0 then the integral of H with respect to B can be defined, and H is said to be B-integrable. Any such process can be approximated by a sequence Hn of left-continuous, adapted and locally bounded processes, in the sense that in probability. Then, the Itô integral is where, again, the limit can be shown to converge in probability. The stochastic integral satisfies the Itô isometry which holds when H is bounded or, more generally, when the integral on the right hand side is finite. Itô processes An Itô process is defined to be an adapted stochastic process that can be expressed as the sum of an integral with respect to Brownian motion and an integral with respect to time, Here, B is a Brownian motion and it is required that σ is a predictable B-integrable process, and μ is predictable and (Lebesgue) integrable. That is, for each t. The stochastic integral can be extended to such Itô processes, This is defined for all locally bounded and predictable integrands. More generally, it is required that Hσ be B-integrable and Hμ be Lebesgue integrable, so that Such predictable processes H are called X-integrable. An important result for the study of Itô processes is Itô's lemma. In its simplest form, for any twice continuously differentiable function f on the reals and Itô process X as described above, it states that f(X) is itself an Itô process satisfying This is the stochastic calculus version of the change of variables formula and chain rule. It differs from the standard result due to the additional term involving the second derivative of f, which comes from the property that Brownian motion has non-zero quadratic variation. Semimartingales as integrators The Itô integral is defined with respect to a semimartingale X. These are processes which can be decomposed as X = M + A for a local martingale M and finite variation process A. Important examples of such processes include Brownian motion, which is a martingale, and Lévy processes. For a left continuous, locally bounded and adapted process H the integral H · X exists, and can be calculated as a limit of Riemann sums. Let πn be a sequence of