Semimartingale

In probability theory, a real valued stochastic process X is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the largest class of processes with respect to which the Itô integral and the Stratonovich integral can be defined. The class of semimartingales is quite large (including, for example, all continuously differentiable processes, Brownian motion and Poisson processes). Submartingales and supermartingales together represent a subset of the semimartingales. A real valued process X defined on the filtered probability space (Ω,F,(Ft)t ≥ 0,P) is called a semimartingale if it can be decomposed as where M is a local martingale and A is a càdlàg adapted process of locally bounded variation. An Rn-valued process X = (X1,...,Xn) is a semimartingale if each of its components Xi is a semimartingale. First, the simple predictable processes are defined to be linear combinations of processes of the form Ht = A1{t > T} for stopping times T and FT -measurable random variables A. The integral H · X for any such simple predictable process H and real valued process X is This is extended to all simple predictable processes by the linearity of H · X in H. A real valued process X is a semimartingale if it is càdlàg, adapted, and for every t ≥ 0, is bounded in probability. The Bichteler-Dellacherie Theorem states that these two definitions are equivalent . Adapted and continuously differentiable processes are continuous finite variation processes, and hence semimartingales. Brownian motion is a semimartingale. All càdlàg martingales, submartingales and supermartingales are semimartingales. Itō processes, which satisfy a stochastic differential equation of the form dX = σdW + μdt are semimartingales. Here, W is a Brownian motion and σ, μ are adapted processes. Every Lévy process is a semimartingale. Although most continuous and adapted processes studied in the literature are semimartingales, this is not always the case.