Английская Википедия: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.

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 <math> Z </math> is of Class D if <math>Z_0=0</math> and the collection

<math> \{Z_T \mid T \text{ a finite-valued stopping time} \} </math>

is uniformly integrable.

The theorem

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

See also

• Doob decomposition theorem

References

• Шаблон:Cite book
• Шаблон:Cite journal
• Шаблон:Cite journal
• Шаблон:Cite book