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 an 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.

Definition

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

X_{t}=M_{t}+A_{t}

where M is a local martingale and A is a càdlàg adapted process of locally bounded variation.

An Rⁿ-valued process X = (X¹,…,Xⁿ) 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_{{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

H\cdot X_{t}\equiv 1_{\{t>T\}}A(X_{t}-X_{T}).

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,

\left\{H\cdot X_{t}:H{\rm {\ is\ simple\ predictable\ and\ }}|H|\leq 1\right\}

is bounded in probability. The Bichteler-Dellacherie Theorem states that these two definitions are equivalent (Protter 2004, p. 144).

Examples

Adapted and continuously differentiable processes are finite variation processes, and hence are 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 measure.
If X is an R^m valued semimartingale and f is a twice continuously differentiable function from R^m to Rⁿ, then f(X) 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.
(Jacod's Countable Expansion) 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. (Protter 2004, p. 53)

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. (Rogers & Williams 1987, p. 358)

For example, if X is an Itō process satisfying the stochastic differential equation dX_t = σ_t dW_t + b_t dt, then

M_{t}=X_{0}+\int _{0}^{t}\sigma _{s}\,dW_{s},\ A_{t}=\int _{0}^{t}b_{s}\,ds.

Special semimartingales

A special semimartingale is a real valued process X with the decomposition X = M + A, where M is a local martingale and A 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 (Protter 2004, p. 130).

For example, every continuous semimartingale is a special semimartingale, in which case M and A are both continuous processes.

Purely discontinuous semimartingales

A semimartingale is called purely discontinuous if its quadratic variation [X] is a pure jump process,

[X]_{t}=\sum _{s\leq t}\Delta X_{s}^{2}

.

Every adapted finite variation process is a purely discontinuous semimartingale. A continuous process is a purely discontinuous semimartingale if and only if it is an adapted finite variation process.

Then, every semimartingale has the unique decomposition X = M + A where M is a continuous local martingale and A is a purely discontinuous semimartingale starting at zero. The local martingale M - M₀ is called the continuous martingale part of X, and written as X^c (He, Wang & Yan 1992, p. 209; Kallenberg 2002, p. 527).

In particular, if X is continuous, then M and A are continuous.

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(X) is a semimartingale for every smooth function f from M to R. (Rogers 1987, p. 24) Stochastic calculus for semimartingales on general manifolds requires the use of the Stratonovich integral.

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References

He, Sheng-wu; Wang, Jia-gang; Yan, Jia-an (1992), Semimartingale Theory and Stochastic Calculus, Science Press, CRC Press Inc., ISBN 0-8493-7715-3
Kallenberg, Olav (2002), Foundations of Modern Probability (2nd ed.), Springer, ISBN 0-387-95313-2
Protter, Philip E. (2004), Stochastic Integration and Differential Equations (2nd ed.), Springer, ISBN 3-540-00313-4
Rogers, L.C.G.; Williams, David (1987), Diffusions, Markov Processes, and Martingales, 2, John Wiley & Sons Ltd, ISBN 0-471-91482-7
Karandikar, Rajeeva L.; Rao, B.V. (2018), Introduction to Stochastic Calculus, Springer Ltd, ISBN 978-981-10-8317-4

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