# 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 $Z$ is of Class D if $Z_0=0$ and the collection

$\{Z_T \mid \text{T a finite valued stopping time} \}$

## The theorem

Let $Z$ be a cadlag submartingale of class D with $Z_0 =0$. 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.[5]

## Notes

1. Doob 1953
2. Meyer 1952
3. Meyer 1963
4. Protter 2005
5. Protter (2005)

## References

• Doob, J. L. (1953). Stochastic Processes. Wiley.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
• Meyer, Paul-André (1962). "A Decomposition theorem for supermartingales". Illinois Journal of Mathematics. 6 (2): 193–205.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
• Meyer, Paul-André (1963). "Decomposition of Supermartingales: the Uniqueness Theorem". Illinois Journal of Mathematics. 7 (1): 1–17.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
• Protter, Philip (2005). Stochastic Integration and Differential Equations. Springer-Verlag. pp. 107–113. ISBN 3-540-00313-4.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>