Semimartingale

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In probability theory, a real valued 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 Itō 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[edit]

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

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[edit]

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

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|\le 1\right\}

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

Examples[edit]

  • 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.

Properties[edit]

  • 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 Rm valued semimartingale and f is a twice continuously differentiable function from Rm 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 Ft, and is adapted with respect to the subfiltration Gt, then X is a Gt-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 Ft is a filtration, and Gt is the filtration generated by Ft and a countable set of disjoint measurable sets. Then, every Ft-semimartingale is also a Gt-semimartingale. (Protter 2004, p. 53)

Semimartingale decompositions[edit]

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

Continuous semimartingales[edit]

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 dXt = σt dWt + bt dt, then

M_t=X_0+\int_0^t\sigma_s\,dW_s,\ A_t=\int_0^t b_s\,ds.

Special semimartingales[edit]

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 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.

Purely discontinuous semimartingales[edit]

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

[X]_t=\sum_{s\le 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 Xc (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[edit]

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.

See also[edit]

References[edit]

  • 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