# 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.^{[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 submartingale is of Class D if and the collection

is uniformly integrable.^{[5]}

## The theorem

Let be a cadlag submartingale of class D with . Then there exists a unique, increasing, predictable process with such that is a uniformly integrable martingale.^{[5]}

## See also

## Notes

## References

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