Doob decomposition theorem
In the theory of in , 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 (or "drift") 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.
Let (Ω, F, ℙ) be a probability space, I = {0, 1, 2, . . . , N} with N ∈ ℕ or I = ℕ0 a finite or an infinite index set, (Fn)n∈I a filtration of F, and X = (Xn)n∈I an adapted stochastic process with E[|Xn|] < ∞ for all n ∈ I. Then there exists a martingale M = (Mn)n∈I and an integrable predictable process A = (An)n∈I starting with A0 = 0 such that Xn = Mn + An for every n ∈ I. Here predictable means that An is Fn−1-measurable for every n ∈ I \ {0}. This decomposition is almost surely unique.
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