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.^[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 ${\displaystyle Z}$ is of Class D if ${\displaystyle Z_{0}=0}$ and the collection

${\displaystyle \{Z_{T}\mid T{\text{ a finite-valued stopping time}}\}}$

is uniformly integrable.^[5]

Theorem

Let ${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} )}$ be a filtered probability space satisfying the usual conditions (i.e. the filtration is right-continuous and complete; see Filtration (probability theory)). If ${\displaystyle X=(X_{t})_{t\geq 0}}$ is a right-continuous submartingale of class D, then there exist unique adapted processes ${\displaystyle M}$ and ${\displaystyle A}$ such that

${\displaystyle X_{t}=M_{t}+A_{t},\qquad t\geq 0,}$

where

• ${\displaystyle M}$ is a uniformly integrable martingale,
• ${\displaystyle A}$ is a predictable, right-continuous, increasing process with ${\displaystyle A_{0}=0}$.

The decomposition ${\displaystyle (M,A)}$ is unique up to indistinguishability.

Remark. For a class D supermartingale, the process A is integrable and of finite variation on bounded intervals.^[6]

See also

• Doob decomposition theorem