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

${\displaystyle X_{t}=M_{t}+A_{t}}$ 

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.

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

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

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,

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

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

• Adapted and continuously differentiable processes are continuous 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.

• 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)

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 dX_t = σ_t dW_t + b_t dt, then

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

Special semimartingales Edit

A special semimartingale is a real valued process ${\displaystyle X}$  with the decomposition ${\displaystyle X=M^{X}+B^{X}}$ , where ${\displaystyle M^{X}}$  is a local martingale and ${\displaystyle B^{X}}$  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.

Multiplicative decompositions Edit

Recall that ${\displaystyle {\mathcal {E}}(X)}$  denotes the stochastic exponential of semimartingale ${\displaystyle X}$ . If ${\displaystyle X}$  is a special semimartingale such that ${\displaystyle \Delta B^{X}\neq -1}$ , then ${\displaystyle {\mathcal {E}}(B^{X})\neq 0}$  and ${\displaystyle {\mathcal {E}}(X)/{\mathcal {E}}(B^{X})={\mathcal {E}}\left(\int _{0}^{\cdot }{\frac {M_{u}^{X}}{1+\Delta B_{u}^{X}}}\right)}$  is a local martingale.^[1] Process ${\displaystyle {\mathcal {E}}(B^{X})}$  is called the multiplicative compensator of ${\displaystyle {\mathcal {E}}(X)}$  and the identity ${\displaystyle {\mathcal {E}}(X)={\mathcal {E}}\left(\int _{0}^{\cdot }{\frac {M_{u}^{X}}{1+\Delta B_{u}^{X}}}\right){\mathcal {E}}(B^{X})}$  the multiplicative decomposition of ${\displaystyle {\mathcal {E}}(X)}$ .

Purely discontinuous semimartingales / quadratic pure-jump semimartingales Edit

A semimartingale is called purely discontinuous (Kallenberg 2002) if its quadratic variation [X] is a finite variation pure-jump process, i.e.,

${\displaystyle [X]_{t}=\sum _{s\leq t}(\Delta X_{s})^{2}}$ .

By this definition, time is a purely discontinuous semimartingale even though it exhibits no jumps at all. Alternative (and preferred) terminology quadratic pure-jump semimartingale (Protter 2004, p. 71) refers to 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 ${\displaystyle X^{c}}$  starting at zero such that ${\displaystyle X-X^{c}}$  is a quadratic pure-jump semimartingale (He, Wang & Yan 1992, p. 209; Kallenberg 2002, p. 527). The local martingale ${\displaystyle X^{c}}$  is called the continuous martingale part of X.

Observe that ${\displaystyle X^{c}}$  is measure-specific. If ${\displaystyle P}$  and ${\displaystyle Q}$  are two equivalent measures then ${\displaystyle X^{c}(P)}$  is typically different from ${\displaystyle X^{c}(Q)}$ , while both ${\displaystyle X-X^{c}(P)}$  and ${\displaystyle X-X^{c}(Q)}$  are quadratic pure-jump semimartingales. By Girsanov's theorem ${\displaystyle X^{c}(P)-X^{c}(Q)}$  is a continuous finite variation process, yielding ${\displaystyle [X^{c}(P)]=[X^{c}(Q)]=[X]-\sum _{s\leq \cdot }(\Delta X_{s})^{2}}$ .

Continuous-time and discrete-time components of a semimartingale Edit

Every semimartingale ${\displaystyle X}$  has a unique decomposition

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) harv error: no target: CITEREFRogers1987 (help) Stochastic calculus for semimartingales on general manifolds requires the use of the Stratonovich integral.

• Sigma-martingale

• 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, vol. 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