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Doob–Meyer decomposition theorem

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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.

Contents

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 submartingale Z is of Class D if Z 0 = 0 and the collection

{ Z T T a finite valued stopping time }

is uniformly integrable.

The theorem

Let Z be a cadlag submartingale of class D. 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.

References

Doob–Meyer decomposition theorem Wikipedia


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