Doob–Meyer decomposition theorem explained

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.[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

Z

is of Class D if

Z0=0

and the collection

\{ZT\midTafinite-valuedstoppingtime\}

is uniformly integrable.[5]

The theorem

Let

Z

be a cadlag supermartingale of class D. Then there exists a unique, non-decreasing, predictable process

A

with

A0=0

such that

Mt=Zt+At

is a uniformly integrable martingale.

See also

References

Notes and References

  1. Doob 1953
  2. Meyer 1952
  3. Meyer 1963
  4. Protter 2005
  5. Protter (2005)

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