Doob decomposition theorem

The Doob decomposition theorem is a result in the theory of discrete time stochastic processes that gives a unique decomposition of any submartingale as the sum of a martingale and an increasing predictable process. The theorem was proved by and is named for J.L. Doob.^[1] The analogous theorem for continuous submartingales is the Doob-Meyer decomposition theorem.

The Theorem

Any submartingale $X_{n}$ has a unique decomposition $X_{n}=M_{n}+A_{n}$ where $M_{n}$ is a martingale and $A_{n}$ is a predictable, increasing process with $A_{0}=0$ .^[2]

References

^ Doob 1953
^ Durrett 2005

Doob, J.L. (1953). Stochastic Processes. Wiley.
Durrett, Rick (2005). Probability: Theory and Examples (3 ed.). Brooks/Cole. p. 234. ISBN 0-534-42441-4.

[1] Doob 1953

[2] Durrett 2005

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