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 a continuous increasing process. It is named for J. L. Doob and Paul-André Meyer.

The theorem

If $X_{t}$ is a continuous submartingale such that the set

$\{X_{\tau }\}$

(where $\tau <\infty$ is a stopping time) is uniformly integrable, then there exists a continuous martingale $M_{t}$ and a continuous increasing process $A_{t}$ such that

$X_{t}=M_{t}+A_{t},\quad \forall t>0$

almost surely.

The processes $M_{t}$ and $A_{t}$ are unique to the point of indistinguishability.

External links

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