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.
History In 1953, Doob published the Doob decomposition theorem which gives a unique decomposition for certain discrete time martingales.Doob 1953 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.Meyer 1962Meyer 1963 In honor of Doob, Meyer used the term "class D" to refer to the class of supermartingales for which his unique decomposition theorem applied.Protter 2005
Class D supermartingales A càdlàg supermartingale is of Class D if and the collection
is uniformly integrable.Protter (2005)
Theorem Let be a filtered probability space satisfying the usual conditions (i.e. the filtration is right-continuous and complete; see Filtration (probability theory)). If is a right-continuous submartingale of class D, then there exist unique adapted processes and such that
where is a uniformly integrable martingale, is a predictable, right-continuous, increasing process with . The decomposition is unique up to indistinguishability.
Remark. For a class D supermartingale, the process A is integrable and of finite variation on bounded intervals.Karatzas & Shreve (1991), Theorem 4.10. See also Doob decomposition theorem
Notes
References