Doob–Meyer decomposition theorem

From formulasearchengine
Jump to navigation Jump to search

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

A càdlàg submartingale is of Class D if and the collection

is uniformly integrable.[5]

The theorem

Let be a cadlag submartingale of class D with . Then there exists a unique, increasing, predictable process with such that is a uniformly integrable martingale.[5]

See also

Notes

  1. Doob 1953
  2. Meyer 1952
  3. Meyer 1963
  4. Protter 2005
  5. 5.0 5.1 Protter (2005)

References

  • {{#invoke:citation/CS1|citation

|CitationClass=book }}

  • {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

  • {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

  • {{#invoke:citation/CS1|citation

|CitationClass=book }}

External links