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

In 1953, Doob published the Doob decomposition theorem which gives a unique decomposition for certain discrete time martingales.Doob 1953 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.Meyer 1962Meyer 1963 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 supermartingale $Z$ is of Class D if $Z_0=0$ and the collection

: $\{Z_T \mid T \text{ a finite-valued stopping time} \}$

is uniformly integrable.Protter (2005)

The theorem

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

Notes

References

{{Cite book| last=Doob | first=J. L. | year=1953 | title=Stochastic Processes | publisher=Wiley }}
{{cite journal |last=Meyer |first=Paul-André |year=1962 |title=A Decomposition theorem for supermartingales |journal=Illinois Journal of Mathematics |volume=6 |issue=2 |pages=193–205 |doi=10.1215/ijm/1255632318 |doi-access=free }}
{{cite journal |last=Meyer |first=Paul-André |year=1963 |title=Decomposition of Supermartingales: the Uniqueness Theorem |journal=Illinois Journal of Mathematics |volume=7 |issue=1 |pages=1–17 |doi=10.1215/ijm/1255637477 |doi-access=free }}
{{Cite book| last=Protter | first=Philip | year=2005 | title=Stochastic Integration and Differential Equations | url=https://archive.org/details/stochasticintegr00prot_960 | url-access=limited | publisher=Springer-Verlag | isbn=3-540-00313-4 |pages = [https://archive.org/details/stochasticintegr00prot_960/page/n120 107]–113 }}

Category:Martingale theory

Category:Theorems in statistics

Category:Theorems in probability theory