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 submartingale $Z$ is of Class D if $Z_0=0$ and the collection

$\{Z_T \mid \text{T a finite valued stopping time} \}$

is uniformly integrable.

The theorem

Let $Z$ be a cadlag submartingale of class D with $Z_0 =0$. Then there exists a unique, increasing, predictable process $A$ with $A_0 =0$ such that $M_t = Z_t - A_t$ is a uniformly integrable martingale.

See also

• Doob decomposition theorem

External links

• The Doob–Meyer decomposition theorem with proof, by Jan van Neerven

• Doob 1953
• Meyer 1952
• Meyer 1963
• Protter 2005
• Protter (2005)