Semimartingales are adapted stochastic processesMathworldPlanetmath which can be used as integrators in the general theory of stochastic integration. Examples of semimartingales include Brownian motionMathworldPlanetmath, all local martingalesPlanetmathPlanetmath, finite variation processes and Levy processesMathworldPlanetmath.

Given a filtered probability space (Ω,,(t)t+,), we consider real-valued stochastic processes Xt with time index t ranging over the nonnegative real numbers. Then, semimartingales have historically been defined as follows.


A semimartingale X is a cadlag adapted process having the decomposition X=M+V for a local martingale M and a finite variation process V.

More recently, the following alternative definition has also become common. For simple predictable integrands ξ, the stochastic integral ξ𝑑X is easily defined for any process X. The following definition characterizes semimartingales as processes for which this integral is well behaved.


A semimartingale X is a cadlag adapted process such that

{0tξ𝑑X:|ξ|1 is simple predictable}

is boundedPlanetmathPlanetmathPlanetmath in probability for each tR+.

Writing ξ for the supremum norm of a process ξ, this definition characterizes semimartingales as processes for which


in probability as n for each t>0, where ξn is any sequence of simple predictable processes satisfying ξn0. This property is necessary and, as it turns out, sufficient for the development of a theory of stochastic integration for which results such as bounded convergence holds.

The equivalence of these two definitions of semimartingales is stated by the Bichteler-Dellacherie theorem.

A stochastic process Xt=(Xt1,Xt2,,Xtn) taking values in n is said to be a semimartingale if Xtk is a semimartingale for each k=1,2,,n.

Title semimartingale
Canonical name Semimartingale
Date of creation 2013-03-22 18:36:38
Last modified on 2013-03-22 18:36:38
Owner gel (22282)
Last modified by gel (22282)
Numerical id 8
Author gel (22282)
Entry type Definition
Classification msc 60G07
Classification msc 60G48
Classification msc 60H05