Doob decomposition theorem

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 (or 'drift') starting at zero. The theorem was proved by and is named for Joseph L. Doob. A n = ∑ k = 1 n ( E [ X k | F k − 1 ] − X k − 1 ) {displaystyle A_{n}=sum _{k=1}^{n}{igl (}mathbb {E} -X_{k-1}{igr )}}     (1) M n = X 0 + ∑ k = 1 n ( X k − E [ X k | F k − 1 ] ) , {displaystyle M_{n}=X_{0}+sum _{k=1}^{n}{igl (}X_{k}-mathbb {E} {igr )},}     (2) 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 (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 < ∞ 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. A real-valued stochastic process X is a submartingale if and only if it has a Doob decomposition into a martingale M and an integrable predictable process A that is almost surely increasing. It is a supermartingale, if and only if A is almost surely decreasing. The theorem is valid word by word also for stochastic processes X taking values in the d-dimensional Euclidean space ℝd or the complex vector space ℂd. This follows from the one-dimensional version by considering the components individually. Using conditional expectations, define the processes A and M, for every n ∈ I, explicitly by