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,(F_t)_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. This means that for almost all ${\displaystyle \omega \in \Omega }$ and all compact intervals ${\displaystyle I\subset [0,\infty )}$, the sample path ${\displaystyle I\ni s\mapsto A_{s}(\omega )}$ is of 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 H_t = A1_{{t > T}} for stopping times T and F_T -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,