Class D supermartingales
A càdlàg supermartingale is of Class D if and the collection
The Doob–Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may be decomposed in a unique way as the sum of a martingale and an increasing predictable process. It is named for Joseph L. Doob and Paul-André Meyer.
In 1953, Doob published the Doob decomposition theorem which gives a unique decomposition for certain discrete time martingales. [1] 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. [2] [3] In honor of Doob, Meyer used the term "class D" to refer to the class of supermartingales for which his unique decomposition theorem applied. [4]
A càdlàg supermartingale is of Class D if and the collection
Let be a cadlag supermartingale of class D. Then there exists a unique, non-decreasing, predictable process with such that is a uniformly integrable martingale. [5]
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