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.

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]

A càdlàg supermartingale ${\displaystyle Z}${\displaystyle Z} is of Class D if ${\displaystyle Z_{0}=0}${\displaystyle Z_{0}=0} and the collection

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

is uniformly integrable.^[5]

Let ${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} )}${\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 ${\displaystyle X=(X_{t})_{t\geq 0}}${\displaystyle X=(X_{t})_{t\geq 0}} is a right-continuous submartingale of class D, then there exist unique adapted processes ${\displaystyle M}${\displaystyle M} and ${\displaystyle A}${\displaystyle A} such that

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

where

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

The decomposition ${\displaystyle (M,A)}${\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]

• Doob decomposition theorem

• Doob, J. L. (1953). Stochastic Processes. Wiley.
• Meyer, Paul-André (1962). "A Decomposition theorem for supermartingales". Illinois Journal of Mathematics. 6 (2): 193–205. doi:10.1215/ijm/1255632318 .
• Meyer, Paul-André (1963). "Decomposition of Supermartingales: the Uniqueness Theorem". Illinois Journal of Mathematics. 7 (1): 1–17. doi:10.1215/ijm/1255637477 .
• Protter, Philip (2005). Stochastic Integration and Differential Equations . Springer-Verlag. pp. 107–113. ISBN 3-540-00313-4.
• Karatzas, Ioannis; Shreve, Steven E. (1991). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics. Vol. 113 (2nd ed.). Springer. ISBN 978-0-387-97655-6.

• Martingale theory
• Theorems in statistics
• Theorems in probability theory