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.

Statement

Let be a probability space, I = {0, 1, 2, ..., N} with N \in \N or I = \N_0 a finite or countably infinite index set, a filtration of \mathcal{F}, and X = (Xn)n∈I an adapted stochastic process with E[ for all n ∈ I . Then there exist 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 -measurable for every n ∈ I \ {0} . This decomposition is almost surely unique.

Remark

The theorem is valid word for word also for stochastic processes X taking values in the d -dimensional Euclidean space \Reals^d or the complex vector space \Complex^d. This follows from the one-dimensional version by considering the components individually.

Proof

Existence

Using conditional expectations, define the processes A and M , for every n ∈ I , explicitly by and where the sums for are empty and defined as zero. Here A adds up the expected increments of X , and M adds up the surprises, i.e., the part of every Xk that is not known one time step before. Due to these definitions, An+1 (if n + 1 ∈ I ) and Mn are n -measurable because the process X is adapted, E[ and E[ because the process X is integrable, and the decomposition is valid for every n ∈ I . The martingale property also follows from the above definition, for every n ∈ I \ {0} .

Uniqueness

To prove uniqueness, let be an additional decomposition. Then the process Y := M − M = A − A is a martingale, implying that and also predictable, implying that for any n ∈ I \ {0} . Since by the convention about the starting point of the predictable processes, this implies iteratively that almost surely for all n ∈ I , hence the decomposition is almost surely unique.

Corollary

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.

Proof

If X is a submartingale, then for all k ∈ I \ {0} , which is equivalent to saying that every term in definition of A is almost surely positive, hence A is almost surely increasing. The equivalence for supermartingales is proved similarly.

Example

Let be a sequence in independent, integrable, real-valued random variables. They are adapted to the filtration generated by the sequence, i.e. for all n ∈ . By and, 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. If the sequence 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.

Application

In mathematical finance, the Doob decomposition theorem can be used to determine the largest optimal exercise time of an American option. Let denote the non-negative, discounted payoffs of an American option in a N -period financial market model, adapted to a filtration (0, 1, . . ., N) , and let \mathbb{Q} denote an equivalent martingale measure. Let denote the Snell envelope of X with respect to \mathbb{Q}. The Snell envelope is the smallest \mathbb{Q} -supermartingale dominating X and in a complete financial market it represents the minimal amount of capital necessary to hedge the American option up to maturity. Let denote the Doob decomposition with respect to \mathbb{Q} of the Snell envelope U into a martingale and a decreasing predictable process with . Then the largest stopping time to exercise the American option in an optimal way is Since A is predictable, the event {τmax = n} = {An = 0, An+1 < 0} is in n for every n ∈ {0, 1,. . ., N − 1} , hence τmax is indeed a stopping time. 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 \mathbb{Q}.

Generalization

The Doob decomposition theorem can be generalized from probability spaces to σ-finite measure spaces.

Citations