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A stochastic process that can be represented as the sum of a local martingale and a process of locally bounded variation. For the formal definition of a semi-martingale one starts from a stochastic basis , where (cf. Stochastic processes, filtering of). A stochastic process is called a semi-martingale if its trajectories are right-continuous and have left limits, and if it can be represented in the form , where is a local martingale and is a process of locally bounded variation, that is,

In general this representation is non-unique. But in the class of representations with predictable processes , the representation is unique (up to stochastic equivalence). The following belong to the class of semi-martingales (apart from local martingales and processes of locally bounded variation): local super-martingales and submartingales, processes with independent increments for which is a function of locally bounded variation for any (and so all processes with stationary independent increments), Itô processes, diffusion-type processes, and others. The class of semi-martingales is invariant under an equivalent change of measure. If is a semi-martingale and is twice continuously differentiable, then is also a semi-martingale. Here (Itô's formula)

or, equivalently,

where is the quadratic variation of the semi-martingale , that is,

is the continuous part of the quadratic variation , , and the integrals are understood as stochastic integrals with respect to a semi-martingale (cf. Stochastic integral).

If is a semi-martingale, then the process with

has bounded jumps, , and so can be uniquely represented as

where is a predictable random process of locally bounded variation and is a local martingale. This martingale can be uniquely represented as , where is a continuous local martingale (a continuous martingale forming the semi-martingale ) and is a purely-discontinuous local martingale that can be written in the form

where is the random jump measure of , that is,

and is its compensator. Since

each semi-martingale admits a representation

called the canonical representation (decomposition).

The set of (predictable) characteristics , where is the quadratic characteristic of , that is, a predictable increasing process such that is a local martingale, is called a triplet of local (predictable) characteristics of .


[1] J. Jacod, "Calcul stochastique et problèmes de martingales" , Lect. notes in math. , 714 , Springer (1979)
[2] R.Sh. Liptser, A.N. [A.N. Shiryaev] Shiryayev, "Theory of martingales" , Kluwer (1989) (Translated from Russian)


See also Itô formula and Stochastic integral. Semi-martingales are the most general stochastic processes with respect to which it is possible to integrate predictable processes in a reasonable way.


[a1] K. Bichteler, "The stochastic integral as a vector measure" , Measure Theory (Oberwolfach, 1979) , Lect. notes in math. , 794 , Springer (1980) pp. 348–360
[a2] C. Dellacherie, "Un survol de la théorie de l'intégrale stochastique" , Measure Theory (Oberwolfach, 1979) , Lect. notes in math. , 794 , Springer (1980) pp. 365–395
[a3] C. Dellacherie, P.A. Meyer, "Probabilités et potentiels" , 2 , Hermann (1980) pp. Chapts. V-VIII: Théorie des martingales
[a4] M. Metivier, "Semimartingales" , de Gruyter (1982)
[a5] L. Schwartz, "Les semi-martingales formelles" , Sem. Probab. XV , Lect. notes in math. , 850 , Springer (1981) pp. 413–489
[a6] J. Jacod, A.N. Shiryaev, "Limit theorems for stochastic processes" , Springer (1987) (Translated from Russian)
How to Cite This Entry:
Semi-martingale. A.N. Shiryaev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098