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. 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. 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. A càdlàg submartingale Z {displaystyle Z} is of Class D if Z 0 = 0 {displaystyle Z_{0}=0} and the collection is uniformly integrable. Let Z {displaystyle Z} be a cadlag submartingale of class D. Then there exists a unique, increasing, predictable process A {displaystyle A} with A 0 = 0 {displaystyle A_{0}=0} such that M t = Z t − A t {displaystyle M_{t}=Z_{t}-A_{t}} is a uniformly integrable martingale.