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.
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.
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 (Protter 2004, p. 144).
Although most continuous and adapted processes studied in the literature are semimartingales, this is not always the case.
By definition, every semimartingale is a sum of a local martingale and a finite-variation process.
(Rogers & Williams 1987, p. 358) For example, if X is an Itō process satisfying the stochastic differential equation dXt = σt dWt + bt dt, then A special semimartingale is a real valued process
is a predictable finite-variation process starting at zero.
If this decomposition exists, then it is unique up to a P-null set.
Conversely, a semimartingale is a special semimartingale if and only if the process Xt* ≡ sups ≤ t |Xs| is locally integrable (Protter 2004, p. 130).
For example, every continuous semimartingale is a special semimartingale, in which case M and A are both continuous processes.
denotes the stochastic exponential of semimartingale
is a special semimartingale such that[clarification needed]
A semimartingale is called purely discontinuous (Kallenberg 2002) if its quadratic variation [X] is a finite-variation pure-jump process, i.e., By this definition, time is a purely discontinuous semimartingale even though it exhibits no jumps at all.
The alternative (and preferred) terminology quadratic pure-jump semimartingale for a purely discontinuous semimartingale (Protter 2004, p. 71) is motivated by the fact that the quadratic variation of a purely discontinuous semimartingale is a pure jump process.
An adapted continuous process is a quadratic pure-jump semimartingale if and only if it is of finite variation.
For every semimartingale X there is a unique continuous local martingale
is a quadratic pure-jump semimartingale (He, Wang & Yan 1992, p. 209; Kallenberg 2002, p. 527).
is called the continuous martingale part of X.
is a continuous finite-variation process, yielding
component does not jump at predictable times, and the
component is equal to the sum of its jumps at predictable times in the semimartingale topology.
The "dp" component is often taken to be a Markov chain but in general the predictable jump times may not be isolated points; for example, in principle
is not necessarily of finite variation, even though it is equal to the sum of its jumps (in the semimartingale topology).
to have independent increments, with jumps at times
The concept of semimartingales, and the associated theory of stochastic calculus, extends to processes taking values in a differentiable manifold.
A process X on the manifold M is a semimartingale if f(X) is a semimartingale for every smooth function f from M to R. (Rogers & Williams 1987, p. 24) Stochastic calculus for semimartingales on general manifolds requires the use of the Stratonovich integral.