Doob–Meyer decomposition theorem

From HandWiki

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 supermartingale [math]\displaystyle{ Z }[/math] is of Class D if [math]\displaystyle{ Z_0=0 }[/math] and the collection

[math]\displaystyle{ \{Z_T \mid T \text{ a finite-valued stopping time} \} }[/math]

is uniformly integrable.[5]

The theorem

Let [math]\displaystyle{ Z }[/math] be a cadlag supermartingale of class D. Then there exists a unique, non-decreasing, predictable process [math]\displaystyle{ A }[/math] with [math]\displaystyle{ A_0 =0 }[/math] such that [math]\displaystyle{ M_t = Z_t + A_t }[/math] 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