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.
a finite or countably infinite index set,
, and X = (Xn)n∈I an adapted stochastic process with E[|Xn|] < ∞ for all n ∈ I.
[2][3][4] The theorem is valid word for word also for stochastic processes X taking values in the d-dimensional Euclidean space
Due to these definitions, An+1 (if n + 1 ∈ I) and Mn are Fn-measurable because the process X is adapted, E[|An|] < ∞ and E[|Mn|] < ∞ because the process X is integrable, and the decomposition Xn = Mn + An is valid for every n ∈ I.
To prove uniqueness, let X = M' + A' be an additional decomposition.
Since Y0 = A'0 − A0 = 0 by the convention about the starting point of the predictable processes, this implies iteratively that Yn = 0 almost surely for all n ∈ I, hence the 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.
The equivalence for supermartingales is proved similarly.
be a sequence in independent, integrable, real-valued random variables.
They are adapted to the filtration generated by the sequence, i.e. Fn = σ(X0, .
By (1) and (2), the Doob decomposition is given by and If the random variables of the original sequence X have mean zero, this simplifies to hence both processes are (possibly time-inhomogeneous) random walks.
consists of symmetric random variables taking the values +1 and −1, then X is bounded, but the martingale M and the predictable process A are unbounded simple random walks (and not uniformly integrable), and Doob's optional stopping theorem might not be applicable to the martingale M unless the stopping time has a finite expectation.
In mathematical finance, the Doob decomposition theorem can be used to determine the largest optimal exercise time of an American option.
, XN) denote the non-negative, discounted payoffs of an American option in a N-period financial market model, adapted to a filtration (F0, F1, .
denote an equivalent martingale measure.
-supermartingale dominating X[8] and in a complete financial market it represents the minimal amount of capital necessary to hedge the American option up to maturity.
[9] Let U = M + A denote the Doob decomposition with respect to
Then the largest stopping time to exercise the American option in an optimal way[10][11] is Since A is predictable, the event {τmax = n} = {An = 0, An+1 < 0} is in Fn for every n ∈ {0, 1, .
It gives the last moment before the discounted value of the American option will drop in expectation; up to time τmax the discounted value process U is a martingale with respect to