# Doob–Meyer decomposition theorem

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. 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. In honor of Doob, Meyer used the term "class D" to refer to the class of supermartingales for which his unique decomposition theorem applied.

## Class D supermartingales

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

$\{Z_{T}\mid T{\text{ a finite-valued stopping time}}\}$ ## The theorem

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