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Doob–Meyer decomposition theorem

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[edit]

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

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

isuniformly integrable.^[5]

The theorem[edit]

Let ${\displaystyle Z}$ be a cadlag supermartingale of class D. Then there exists a unique, non-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[edit]

• Doob decomposition theorem

Notes[edit]

References[edit]

• 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.