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 ${\displaystyle Z}$ is of Class D if ${\displaystyle Z_{0}=0}$ and the collection

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

is uniformly integrable.^[5]

The theorem

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

See also

• Doob decomposition theorem

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. doi:10.1215/ijm/1255632318.
• Meyer, Paul-André (1963). "Decomposition of Supermartingales: the Uniqueness Theorem". Illinois Journal of Mathematics. 7 (1): 1–17. doi:10.1215/ijm/1255637477.
• Protter, Philip (2005). Stochastic Integration and Differential Equations. Springer-Verlag. pp. 107–113. ISBN 3-540-00313-4.