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In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale and a predictable process starting at zero. The theorem was proved by and is named for Joseph L. Doob.
The analogous theorem in the continuous-time case is the Doob–Meyer decomposition theorem.
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