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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.^[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]

Class D supermartingales

A càdlàg supermartingale $Z$ is of Class D if $Z_{0}=0$ and the collection

\{Z_{T}\mid T{\text{ a finite-valued stopping time}}\}

is uniformly integrable.^[5]

The theorem

Let $Z$ be a cadlag supermartingale of class D. Then there exists a unique, non-decreasing, predictable process $A$ with $A_{0}=0$ such that $M_{t}=Z_{t}+A_{t}$ is a uniformly integrable martingale.^[5]