partitions of [0, t] with mesh going to zero, This limit converges in probability. The stochastic integral of left-continuous processes is general enough for studying much of stochastic calculus. For example, it is sufficient for applications of Itô's Lemma, changes of measure via Girsanov's theorem, and for the study of stochastic differential equations. However, it is inadequate for other important topics such as martingale representation theorems and local times. The integral extends to all predictable and locally bounded integrands, in a unique way, such that the dominated convergence theorem holds. That is, if Hn → ;H and |Hn| ≤ J for a locally bounded process J, then in probability. The uniqueness of the extension from left-continuous to predictable integrands is a result of the monotone class lemma. In general, the stochastic integral H · X can be defined even in cases where the predictable process H is not locally bounded. If K = 1 / (1 + |H|) then K and KH are bounded. Associativity of stochastic integration implies that H is X-integrable, with integral H · X = Y, if and only if Y0 = 0 and K · Y = (KH) · X. The set of X-integrable processes is denoted by L(X). Properties The following properties can be found in works such as and : The stochastic integral is a càdlàg process. Furthermore, it is a semimartingale. The discontinuities of the stochastic integral are given by the jumps of the integrator multiplied by the integrand. The jump of a càdlàg process at a time t is Xt − Xt−, and is often denoted by ΔXt. With this notation, Δ(H · X) = H ΔX. A particular consequence of this is that integrals with respect to a continuous process are always themselves continuous. Associativity. Let J, K be predictable processes, and K be X-integrable. Then, J is K · X integrable if and only if JK is X integrable, in which case Dominated convergence. Suppose that Hn → H and |Hn| ≤ J, where J is an X-integrable process. then Hn · X → H · X. Convergence is in probability at each time t. In fact, it converges uniformly on compact sets in probability. The stochastic integral commutes with the operation of taking quadratic covariations. If X and Y are semimartingales then any X-integrable process will also be [X, Y]-integrable, and [H · X, Y] = H ·