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. Submartingales and supermartingales together represent a subset of the semimartingales.

Definition

A real-valued process X defined on the filtered probability space 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 and all compact intervals, the sample path is of bounded variation.
An Rⁿ-valued process X = is a semimartingale if each of its components Xⁱ is a semimartingale.

Alternative definition

First, the simple predictable processes are defined to be linear combinations of processes of the form H_t = A1 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,
is bounded in probability. The Bichteler–Dellacherie Theorem states that these two definitions are equivalent.

Examples

Adapted and continuously differentiable processes are continuous, locally 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.

Fractional Brownian motion with Hurst parameter H ≠ 1/2 is not a semimartingale.
Properties
The semimartingales form the largest class of processes for which the Itō integral can be defined.
Linear combinations of semimartingales are semimartingales.
Products of semimartingales are semimartingales, which is a consequence of the integration by parts formula for the Itō integral.
The quadratic variation exists for every semimartingale.
The class of semimartingales is closed under optional stopping, localization, change of time and absolutely continuous change of probability measure.
If X is an R^m-valued semimartingale and f is a twice continuously differentiable function from R^m to Rⁿ, then f is a semimartingale. This is a consequence of Itō's lemma.
The property of being a semimartingale is preserved under shrinking the filtration. More precisely, if X is a semimartingale with respect to the filtration F_t, and is adapted with respect to the subfiltration G_t, then X is a G_t-semimartingale.
The property of being a semimartingale is preserved under enlarging the filtration by a countable set of disjoint sets. Suppose that F_t is a filtration, and G_t is the filtration generated by F_t and a countable set of disjoint measurable sets. Then, every F_t-semimartingale is also a G_t-semimartingale.
Semimartingale decompositions

By definition, every semimartingale is a sum of a local martingale and a finite-variation process. However, this decomposition is not unique.

Continuous semimartingales

A continuous semimartingale uniquely decomposes as X = M + A where M is a continuous local martingale and A is a continuous finite-variation process starting at zero.
For example, if X is an Itō process satisfying the stochastic differential equation dX_t = σ_t dW_t + b_t dt, then

Special semimartingales

A special semimartingale is a real-valued process with the decomposition, where is a local martingale and is a predictable finite-variation process starting at zero. If this decomposition exists, then it is unique up to a P-null set.
Every special semimartingale is a semimartingale. Conversely, a semimartingale is a special semimartingale if and only if the process X_t^* ≡ sup_{s ≤ t} |X_s| is locally integrable.
For example, every continuous semimartingale is a special semimartingale, in which case M and A are both continuous processes.

Multiplicative decompositions

Recall that denotes the stochastic exponential of semimartingale. If is a special semimartingale such that, then and is a local martingale. Process is called the multiplicative compensator of and the identity the multiplicative decomposition of.

Purely discontinuous semimartingales / quadratic pure-jump semimartingales

A semimartingale is called purely discontinuous if its quadratic variation 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 terminology quadratic pure-jump semimartingale for a purely discontinuous semimartingale is motivated by the fact that the quadratic variation of a purely discontinuous semimartingale is a pure jump process. Every finite-variation semimartingale is a quadratic pure-jump semimartingale. 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 starting at zero such that is a quadratic pure-jump semimartingale. The local martingale is called the continuous martingale part of ''X''.
Observe that is measure-specific. If ' and ' are two equivalent measures then is typically different from, while both and are quadratic pure-jump semimartingales. By Girsanov's theorem is a continuous finite-variation process, yielding.

Continuous-time and discrete-time components of a semimartingale

Every semimartingale has a unique decomposition where, the 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. One then has. Typical examples of the "qc" component are Itô process and Lévy process. 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 may jump at every rational time. Observe also that is not necessarily of finite variation, even though it is equal to the sum of its jumps. For example, on the time interval take to have independent increments, with jumps at times taking values with equal probability.

Canonical Decomposition

Although the decomposition of a semimartingale into the sum of a local martingale and a finite-variation process is not unique, every semimartingale has a unique canonical decomposition asfor a unique local martingale and a unique predictable finite variation process starting at zero.

Semimartingales on a manifold

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 is a semimartingale for every smooth function f from M to R. Stochastic calculus for semimartingales on general manifolds requires the use of the Stratonovich integral.