# 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.

## Contents

## 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

- Doob, J. L. (1953).
*Stochastic Processes*. Wiley. - Meyer, Paul-André (1962). "A Decomposition theorem for supermartingales".
*Illinois Journal of Mathematics*.**6**(2): 193–205. - Meyer, Paul-André (1963). "Decomposition of Supermartingales: the Uniqueness Theorem".
*Illinois Journal of Mathematics*.**7**(1): 1–17. - Protter, Philip (2005).
*Stochastic Integration and Differential Equations*. Springer-Verlag. pp. 107–113. ISBN 3-540-00313-4.