The Doob–Meyer decomposition theorem is a theorem in
stochastic calculus
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...
stating the conditions under which a
submartingale
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all ...
may be decomposed in a unique way as the sum of a
martingale and an
increasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
predictable process In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose value is knowable at a prior time. The predictable processes form the smallest class that is closed under taking limits of ...
. It is named for
Joseph L. Doob and
Paul-André Meyer
Paul-André Meyer (21 August 1934 – 30 January 2003) was a French mathematician, who played a major role in the development of the general theory of stochastic processes.
He worked at the Institut de Recherche Mathématique (IRMA) in Str ...
.
History
In 1953, Doob published the
Doob decomposition theorem
In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale an ...
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
Paul-André Meyer (21 August 1934 – 30 January 2003) was a French mathematician, who played a major role in the development of the general theory of stochastic processes.
He worked at the Institut de Recherche Mathématique (IRMA) in Str ...
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.
[Protter 2005]
Class D supermartingales
A
càdlàg In mathematics, a càdlàg (French: "''continue à droite, limite à gauche''"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset ...
supermartingale is of Class D if
and the collection
:
is
uniformly integrable.
[Protter (2005)]
The theorem
Let
be a cadlag
supermartingale of class D. Then there exists a unique, increasing,
predictable process In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose value is knowable at a prior time. The predictable processes form the smallest class that is closed under taking limits of ...
with
such that
is a uniformly integrable martingale.
See also
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Doob decomposition theorem
In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale an ...
Notes
References
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{{DEFAULTSORT:Doob-Meyer decomposition theorem
Martingale theory
Theorems in statistics
Probability theorems