Semimartingale

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. This means that for almost all $\backslash omega\; \backslash in\; \backslash Omega$ and all compact intervals $I\; \backslash subset\; [0,\backslash infty)$, the sample path $I\; \backslash ni\; s\; \backslash mapsto\; A\_s(\backslash omega)$ is of 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

:$H\backslash cdot\; X\_t\; :=\; 1\_\{\backslash \{t>T\backslash \}\}A(X\_t-X\_T).$

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,

:$\backslash left\backslash \{H\backslash cdot\; X\_t:H\{\backslash rm\backslash \; is\backslash \; simple\backslash \; predictable\backslash \; and\backslash \; \}|H|\backslash le\; 1\backslash right\backslash \}$

is bounded in probability. The Bichteler–Dellacherie Theorem states that these two definitions are equivalent {{Harv|Protter|2004|p=144}}.

• 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.

• 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 (see Girsanov's Theorem).
• 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. {{Harv|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.

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. {{Harv|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

:$M\_t=X\_0+\backslash int\_0^t\backslash sigma\_s\backslash ,dW\_s,\backslash \; A\_t=\backslash int\_0^t\; b\_s\backslash ,ds.$

A special semimartingale is a real valued process $X$ with the decomposition $X\; =\; M^X\; +B^X$, where $M^X$ is a local martingale and $B^X$ 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 {{Harv|Protter|2004|p=130}}.

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

Recall that $\backslash mathcal\{E\}(X)$ denotes the stochastic exponential of semimartingale $X$. If $X$ is a special semimartingale such that{{what|reason=What is Delta?|date=October 2023}} $\backslash Delta\; B^X\; \backslash neq\; -1$, then $\backslash mathcal\{E\}(B^X)\backslash neq\; 0$ and $\backslash mathcal\{E\}(X)/\backslash mathcal\{E\}(B^X)=\backslash mathcal\{E\}\backslash left(\backslash int\_0^\backslash cdot\; \backslash frac\{M^X\_u\}\{1+\backslash Delta\; B^X\_u\}\backslash right)$ is a local martingale.{{Cite journal|last1=Lépingle|first1=Dominique|last2=Mémin|first2=Jean|date=1978|title=Sur l'integrabilité uniforme des martingales exponentielles|url=|journal=Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete|language=fr|volume=42|issue=3|at=Proposition II.1|doi=10.1007/BF00641409|issn=0044-3719|doi-access=free}} Process $\backslash mathcal\{E\}(B^X)$ is called the multiplicative compensator of $\backslash mathcal\{E\}(X)$ and the identity $\backslash mathcal\{E\}(X)=\backslash mathcal\{E\}\backslash left(\backslash int\_0^\backslash cdot\; \backslash frac\{M^X\_u\}\{1+\backslash Delta\; B^X\_u\}\backslash right)\backslash mathcal\{E\}(B^X)$ the multiplicative decomposition of $\backslash mathcal\{E\}(X)$.

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

:$[X]\_t=\backslash sum\_\{s\backslash le\; t\}(\backslash Delta\; X\_s)^2$.

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 {{Harv|Protter|2004|p=71}} 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 $X^c$ starting at zero such that $X-X^c$ is a quadratic pure-jump semimartingale ({{Harvnb|He|Wang|Yan|1992|p=209}}; {{Harvnb|Kallenberg|2002|p=527}}). The local martingale $X^c$ is called the continuous martingale part of X.

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

Every semimartingale $X$ has a unique decomposition $$X\; =\; X\_0\; +\; X^\{\backslash mathrm\{qc\}\}\; +X^\{\backslash mathrm\{dp\}\},$$where $X^\{\backslash mathrm\{qc\}\}\_0=X^\{\backslash mathrm\{dp\}\}\_0=0$, the $X^\{\backslash mathrm\{qc\}\}$ component does not jump at predictable times, and the $X^\{\backslash mathrm\{dp\}\}$ component is equal to the sum of its jumps at predictable times in the semimartingale topology. One then has $[X^\{\backslash mathrm\{qc\}\},X^\{\backslash mathrm\{dp\}\}]=0$.{{Cite journal|last1=Černý|first1=Aleš|last2=Ruf|first2=Johannes|date=2021-11-01|title=Pure-jump semimartingales|url=https://projecteuclid.org/journals/bernoulli/volume-27/issue-4/Pure-jump-semimartingales/10.3150/21-BEJ1325.full|journal=Bernoulli|volume=27|issue=4|pages=2631|doi=10.3150/21-BEJ1325|s2cid=202538473 |issn=1350-7265|arxiv=1909.03020}} 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 $X^\{\backslash mathrm\{dp\}\}$ may jump at every rational time. Observe also that $X^\{\backslash mathrm\{dp\}\}$ is not necessarily of finite variation, even though it is equal to the sum of its jumps (in the semimartingale topology). For example, on the time interval $[0,\backslash infty)$ take $X^\{\backslash mathrm\{dp\}\}$ to have independent increments, with jumps at times $\backslash \{\backslash tau\_n\; =\; 2-1/n\backslash \}\_\{n\backslash in\backslash mathbb\{N\}\}$ taking values $\backslash pm\; 1/n$ with equal probability.

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. {{Harv|Rogers|Williams|1987|p=24}} Stochastic calculus for semimartingales on general manifolds requires the use of the Stratonovich integral.

• Sigma-martingale

{{Reflist}}

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{{Stochastic processes}}

Category:Martingale theory