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[edit]
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]
A càdlàg supermartingale is of Class D if and the collection
is uniformly integrable.^{[5]}
The theorem[edit]
Let be a cadlag supermartingale of class D. Then there exists a unique, increasing, predictable process with such that is a uniformly integrable martingale.^{[5]}
See also[edit]
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.
- 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.