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| The '''Doob–Meyer decomposition theorem''' is a theorem in [[stochastic calculus]] stating the conditions under which a [[Martingale (probability theory)#Submartingales and supermartingales|submartingale]] may be decomposed in a unique way as the sum of a [[Martingale (probability theory)|martingale]] and an [[increasing process|increasing]] [[predictable process]]. It is named for [[Joseph L. Doob]] and [[Paul-André Meyer]].
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| ==History==
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| In 1953, Doob published the [[Doob decomposition theorem]] which gives a unique decomposition for certain discrete time martingales.<ref>Doob 1953</ref> He conjectured a continuous time version of the theorem and in two publications in 1962 and 1963 [[Paul-André Meyer]] proved such a theorem, which became known as the Doob-Meyer decomposition.<ref>Meyer 1952</ref><ref>Meyer 1963</ref> In honor of Doob, Meyer used the term "class D" to refer to the class of supermartingales for which his unique decomposition theorem applied.<ref>Protter 2005</ref>
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| ==Class D Supermartingales==
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| A [[càdlàg]] [[Martingale_(probability_theory)#Submartingales_and_supermartingales|submartingale]] <math> Z </math> is of Class D if <math>Z_0=0</math> and the collection
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| :<math> \{Z_T \mid \text{T a finite valued stopping time} \} </math>
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| is [[uniform integrability|uniformly integrable]].<ref name="Protter">Protter (2005)</ref> | |
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| == The theorem ==
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| Let <math>Z</math> be a cadlag [[submartingale]] of class D with <math> Z_0 =0</math>. Then there exists a unique, increasing, [[predictable process]] <math> A</math> with <math> A_0 =0</math> such that <math>M_t = Z_t - A_t</math> is a uniformly integrable martingale.<ref name="Protter" />
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| ==See also==
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| *[[Doob decomposition theorem]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{Cite book| last=Doob | first=J. L. | year=1953 | title=Stochastic Processes | publisher=Wiley | isbn= }}
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| *{{cite journal |last=Meyer |first=Paul |authorlink= |coauthors= |year=1962 |month= |title=A Decomposition theorem for supermartingales |journal=Illinois Journal of Mathematics |volume=6 |issue= |pages=193–205 |id= |url= |accessdate= |quote= }}
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| *{{cite journal |last=Meyer |first=Paul |authorlink= |coauthors= |year=1963 |month= |title=Decomposition of supermartingales: the uniqueness theorem |journal=Illinois Journal of Mathematics |volume=7 |issue= |pages=1–17 |id= |url= |accessdate= |quote= }}
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| *{{Cite book| last=Protter | first=Philip | year=2005 | title=Stochastic Integration and Differential Equations | publisher=Springer-Verlag | isbn=3-540-00313-4 |pages = 107–113 }}
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| == External links ==
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| *[http://fa.its.tudelft.nl/seminar/seminar2003_2004/lecture3.pdf The Doob–Meyer decomposition theorem with proof, by Jan van Neerven]
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| {{DEFAULTSORT:Doob-Meyer decomposition theorem}}
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| [[Category:Martingale theory]]
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| [[Category:Statistical theorems]]
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| [[Category:Probability theorems]]
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