Semimartingale

 ( Martingale Theory ) 

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 (including, for example, all continuously differentiable processes, Wiener process and ). Submartingales and supermartingales together represent a subset of the semimartingales.

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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. 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 R ⁿ-valued process X = ( X¹,..., X ⁿ) is a semimartingale if each of its components X ⁱ is a semimartingale.

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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

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

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 .

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Examples

• Adapted and continuously differentiable processes are continuous, locally finite-variation processes, and hence semimartingales.
• Wiener process 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.

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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 stopped process, 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 R ⁿ, 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.

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

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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 d X_t = σ_t d W_t + b_t dt, then

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

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Special semimartingales 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 .

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

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Multiplicative decompositions Recall that $\backslash mathcal\{E\}(X)$ denotes the stochastic exponential of semimartingale $X$. If $X$ is a special semimartingale such that $\backslash Delta\; B^X\; \backslash neq\; -1$, 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. 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 $\backslash mathcal\{E\}(X)$.

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Purely discontinuous semimartingales / quadratic pure-jump semimartingales 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 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 (; ). 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 theorem $X^c(P)-X^c(Q)$ is a continuous finite-variation process, yielding $X^c(P)=X^c(Q)\; =\; X-\backslash sum\_\{s\backslash leq\backslash cdot\}(\backslash Delta\; X\_s)^2$.

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Continuous-time and discrete-time components of a semimartingale 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 $X^\{\backslash mathrm\{qc\}\},X^\{\backslash mathrm\{dp\}\}=0$. 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.

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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 as$$X\; =\; N^\{X\}+A^\{X\}+\backslash sum\_\{s\backslash leq\backslash cdot\}\backslash mathbf\{1\}\_\{\backslash \{\backslash vert\{\backslash Delta\; X\}\_\{s\}\backslash vert>1\backslash \}\}\backslash Delta\; X\_\{s\},$$for a unique local martingale $N^\{X\}$ and a unique predictable finite variation process $A^\{X\}$ starting at zero.

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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. Stochastic calculus for semimartingales on general manifolds requires the use of the Stratonovich integral.

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See also

• Sigma-martingale

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