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. 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. In honor of Doob, Meyer used the term "class D" to refer to the class of supermartingales for which his unique decomposition theorem applied.
Class D supermartingales A càdlàg supermartingale Z {\displaystyle Z} is of Class D if Z 0 = 0 {\displaystyle Z_{0}=0} and the collection
{ Z T ∣ T a finite-valued stopping time } {\displaystyle \{Z_{T}\mid T{\text{ a finite-valued stopping time}}\}}
is uniformly integrable.
Theorem Let ( Ω , F , ( F t ) t ≥ 0 , 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 X = ( X t ) t ≥ 0 {\displaystyle X=(X_{t})_{t\geq 0}} is a right-continuous submartingale of class D, then there exist unique adapted processes M {\displaystyle M} and A {\displaystyle A} such that
X t = M t + A t , t ≥ 0 , {\displaystyle X_{t}=M_{t}+A_{t},\qquad t\geq 0,}
where
M {\displaystyle M} is a uniformly integrable martingale,
A {\displaystyle A} is a predictable, right-continuous, increasing process with A 0 = 0 {\displaystyle A_{0}=0} . The decomposition ( 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.
See also Doob decomposition theorem
Notes
References 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.