Doob decomposition theorem

In the theory of discrete time stochastic processes, a part of the mathematical theory of probability, the Doob decomposition theorem 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 A₀ = 0.^[2]

References

^ Doob, J.L. (1953). Stochastic Processes. Wiley. ISBN 978-0-471-21813-5. {{cite book}}: ISBN / Date incompatibility (help)
^ Durrett, Rick (2005). Probability: Theory and Examples (3 ed.). Brooks/Cole. p. 234. ISBN 0-534-42441-4.

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[1] Doob, J.L. (1953). Stochastic Processes. Wiley. ISBN 978-0-471-21813-5. {{cite book}}: ISBN / Date incompatibility (help)

[2] Durrett, Rick (2005). Probability: Theory and Examples (3 ed.). Brooks/Cole. p. 234. ISBN 0-534-42441-4.

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