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| In probability theory, a real valued [[stochastic process|process]] ''X'' is called a '''semimartingale''' if it can be decomposed as the sum of a [[local martingale]] and an adapted finite-variation process.
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| Semimartingales are "good integrators", forming the largest class of processes with respect to which the [[Itō integral]] and the [[Stratonovich integral]] can be defined.
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| The class of semimartingales is quite large (including, for example, all continuously differentiable processes, [[Wiener process|Brownian motion]] and [[Poisson process]]es). [[Martingale_(probability_theory)#Submartingales_and_supermartingales|Submartingales]] and [[Martingale_(probability_theory)#Submartingales_and_supermartingales|supermartingales]] together represent a subset of the semimartingales.
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| ==Definition==
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| A real valued process ''X'' defined on the [[filtration (mathematics)#Measure theory|filtered probability space]] (Ω,''F'',(''F''<sub>''t''</sub>)<sub>''t'' ≥ 0</sub>,P) is called a '''semimartingale''' if it can be decomposed as
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| :<math>X_t = M_t + A_t</math> | |
| where ''M'' is a [[local martingale]] and ''A'' is a [[càdlàg]] [[adapted process]] of locally [[bounded variation]].
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| An '''R'''<sup>''n''</sup>-valued process ''X'' = (''X''<sup>1</sup>,…,''X''<sup>''n''</sup>) is a semimartingale if each of its components ''X''<sup>''i''</sup> is a semimartingale.
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| ==Alternative definition==
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| First, the simple [[predictable process]]es are defined to be linear combinations of processes of the form ''H''<sub>''t''</sub> = ''A''1<sub>{''t'' > ''T''}</sub> for stopping times ''T'' and ''F''<sub>''T''</sub> -measurable random variables ''A''. The integral ''H'' · ''X'' for any such simple predictable process ''H'' and real valued process ''X'' is
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| :<math>H\cdot X_t\equiv 1_{\{t>T\}}A(X_t-X_T).</math>
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| This is extended to all simple predictable processes by the linearity of ''H'' · ''X'' in ''H''.
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| A real valued process ''X'' is a semimartingale if it is càdlàg, adapted, and for every ''t'' ≥ 0,
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| :<math>\left\{H\cdot X_t:H{\rm\ is\ simple\ predictable\ and\ }|H|\le 1\right\}</math>
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| is bounded in probability. The Bichteler-Dellacherie Theorem states that these two definitions are equivalent {{Harv|Protter|2004|p=144}}.
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| ==Examples==
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| * Adapted and continuously differentiable processes are finite variation processes, and hence are semimartingales.
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| * [[Wiener process|Brownian motion]] is a semimartingale.
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| * All càdlàg [[Martingale (probability theory)|martingales]], submartingales and supermartingales are semimartingales.
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| * [[Itō calculus#Itō processes|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.
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| * Every [[Lévy process]] is a semimartingale.
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| Although most continuous and adapted processes studied in the literature are semimartingales, this is not always the case.
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| * [[Fractional Brownian motion]] with Hurst parameter ''H'' ≠ 1/2 is not a semimartingale.
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| ==Properties==
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| * The semimartingales form the largest class of processes for which the [[Itō calculus|Itō integral]] can be defined.
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| * Linear combinations of semimartingales are semimartingales.
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| * Products of semimartingales are semimartingales, which is a consequence of the integration by parts formula for the [[Stochastic calculus|Itō integral]].
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| * The [[quadratic variation]] exists for every semimartingale.
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| * The class of semimartingales is closed under [[stopped process|optional stopping]], [[Stopping time#Localization|localization]], [[change of time]] and absolutely continuous [[Absolutely_continuous#Absolute_continuity_of_measures|change of measure]].
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| * If ''X'' is an '''R'''<sup>''m''</sup> valued semimartingale and ''f'' is a twice continuously differentiable function from '''R'''<sup>''m''</sup> to '''R'''<sup>''n''</sup>, then ''f''(''X'') is a semimartingale. This is a consequence of [[Itō's lemma]].
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| * 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''<sub>t</sub>, and is adapted with respect to the subfiltration ''G''<sub>t</sub>, then ''X'' is a ''G''<sub>t</sub>-semimartingale.
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| * (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''<sub>t</sub> is a filtration, and ''G''<sub>t</sub> is the filtration generated by ''F''<sub>t</sub> and a countable set of disjoint measurable sets. Then, every ''F''<sub>t</sub>-semimartingale is also a ''G''<sub>t</sub>-semimartingale. {{Harv|Protter|2004|p=53}}
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| ==Semimartingale decompositions==
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| 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===
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| 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}}
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| For example, if ''X'' is an Itō process satisfying the stochastic differential equation d''X''<sub>t</sub> = σ<sub>t</sub> d''W''<sub>t</sub> + ''b''<sub>t</sub> dt, then
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| :<math>M_t=X_0+\int_0^t\sigma_s\,dW_s,\ A_t=\int_0^t b_s\,ds.</math>
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| ===Special semimartingales===
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| 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.
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| Every special semimartingale is a semimartingale. Conversely, a semimartingale is a special semimartingale if and only if the process ''X''<sub>t</sub><sup>*</sup> ≡ sup<sub>''s'' ≤ ''t''</sub> |X<sub>''s''</sub>| is [[Stopping time#Localization|locally integrable]] {{Harv|Protter|2004|p=130}}.
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| For example, every continuous semimartingale is a special semimartingale, in which case ''M'' and ''A'' are both continuous processes.
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| ===Purely discontinuous semimartingales===
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| A semimartingale is called purely discontinuous if its quadratic variation [''X''] is a pure jump process,
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| :<math>[X]_t=\sum_{s\le t}\Delta X_s^2</math>. | |
| 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.
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| 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'' - ''M''<sub>0</sub> is called the continuous martingale part of ''X'', and written as ''X''<sup>c</sup> ({{Harvnb|He|Wang|Yan|1992|p=209}}; {{Harvnb|Kallenberg|2002|p=527}}).
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| In particular, if ''X'' is continuous, then ''M'' and ''A'' are continuous.
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| ==Semimartingales on a manifold== | |
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| 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|1987|p=24}} Stochastic calculus for semimartingales on general manifolds requires the use of the [[Stratonovich integral]].
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| ==See also==
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| *[[Sigma-martingale]]
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| ==References==
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| *{{Citation|last=He|first=Sheng-wu|last2=Wang|first2=Jia-gang|last3=Yan|first3=Jia-an|year=1992|title=Semimartingale Theory and Stochastic Calculus|publisher=Science Press, CRC Press Inc.|isbn= 0-8493-7715-3}}
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| *{{Citation|last=Kallenberg|first=Olav|year=2002|title=Foundations of Modern Probability|edition=2nd|publisher=Springer|isbn=0-387-95313-2}}
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| *{{Citation|last=Protter|first=Philip E.|year=2004|title=Stochastic Integration and Differential Equations|publisher=Springer|edition=2nd|isbn=3-540-00313-4}}
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| *{{Citation|last=Rogers|first=L.C.G.|last2=Williams|first2=David|year=1987|volume=2|title=Diffusions, Markov Processes, and Martingales|publisher=John Wiley & Sons Ltd|isbn=0-471-91482-7}}
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| {{Stochastic processes}}
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| [[Category:Stochastic processes]]
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| [[Category:Martingale theory]]
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