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. 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 $$ Z $$ is of Class D if $$Z_0=0$$ and the collection
 * $$ \{Z_T \mid T \text{ a finite-valued stopping time} \} $$

is uniformly integrable.

The theorem
Let $$Z$$ be a cadlag supermartingale of class D. Then there exists a unique, non-decreasing, predictable process $$ A$$ with $$ A_0 =0$$ such that $$M_t = Z_t + A_t$$ is a uniformly integrable martingale.