Doob-Meyer decomposition theorem

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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.

[edit] The theorem

If $X t$ is a continuous submartingale such that the set

{X τ}

(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