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. 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. In honor of Doob, Meyer used the term "class D" to refer to the class of supermartingales for which his unique decomposition theorem applied.

Class D supermartingales

A càdlàg supermartingale $Z$ is of Class D if $Z_0=0$ and the collection

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

is uniformly integrable.

The theorem

Let $Z$ be a cadlag supermartingale of class D. Then there exists a unique, non-decreasing, predictable process $A$ with $A_0=0$ such that $M_t=Z_t+A_t$ is a uniformly integrable martingale.

See also

• Doob decomposition theorem

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.