# 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.
The analogous theorem in the continuous-time case is the Doob–Meyer decomposition theorem.

## Statement

Let $(\Omega ,{\mathcal {F}},\mathbb {P} )$ be a probability space, I = {0, 1, 2, ..., N} with $N\in \mathbb {N}$ or $I=\mathbb {N} _{0}$ a finite or countably infinite index set, $({\mathcal {F}}_{n})_{n\in I}$ a filtration of ${\mathcal {F}}$, and X = (X$_n$)$_{n∈I}$ an adapted stochastic process with E[|X$_n$|] < ∞ for all n ∈ I. Then there exist a martingale M = (M$_n$)$_{n∈I}$ and an integrable predictable process A = (A$_n$)$_{n∈I}$ starting with A$_0$ = 0 such that X$_n$ = M$_n$ + A$_n$ for every n ∈ I. Here predictable means that A$_n$ is ${\mathcal {F}}_{n-1}$-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 $\mathbb {R} ^{d}$ or the complex vector space $\mathbb {C} ^{d}$. 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
| $A_{n}=\sum _{k=1}^{n}{\bigl\mathbb {E} [X_{k}\,|\,{\mathcal {F}}_{k-1}]-X_{k-1}{\bigr )}$ | (1) |
and
| $M_{n}=X_{0}+\sum _{k=1}^{n}{\biglX_{k}-\mathbb {E} [X_{k}\,|\,{\mathcal {F}}_{k-1}]{\bigr )},$ | (2) |
where the sums for n = 0 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 X$_k$ that is not known one time step before. Due to these definitions, A$_{n+1}$ (if n + 1 ∈ I) and M$_n$ are F$_n$-measurable because the process X is adapted, E[|A$_n$|] < ∞ and E[|M$_n$|] < ∞ because the process X is integrable, and the decomposition X$_n$ = M$_n$ + A$_n$ is valid for every n ∈ I. The martingale property
    $\mathbb {E} [M_{n}-M_{n-1}\,|\,{\mathcal {F}}_{n-1}]=0$ a.s.
also follows from the above definition (2), for every n ∈ I \ {0}.

### Uniqueness

To prove uniqueness, let X = M' + A' be an additional decomposition. Then the process Y := M − M' = A' − A is a martingale, implying that
    $\mathbb {E} [Y_{n}\,|\,{\mathcal {F}}_{n-1}]=Y_{n-1}$ a.s.,
and also predictable, implying that
    $\mathbb {E} [Y_{n}\,|\,{\mathcal {F}}_{n-1}]=Y_{n}$ a.s.
for any n ∈ I \ {0}. Since Y$_0$ = A'$_0$ − A$_0$ = 0 by the convention about the starting point of the predictable processes, this implies iteratively that Y$_n$ = 0 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
    $\mathbb {E} [X_{k}\,|\,{\mathcal {F}}_{k-1}]\geq X_{k-1}$ a.s.
for all k ∈ I \ {0}, which is equivalent to saying that every term in definition (1) of A is almost surely positive, hence A is almost surely increasing. The equivalence for supermartingales is proved similarly.

## Example

Let X = (X$_n$)$_{n∈N0{\displaystyle\mathbb{N}_{0}}}$ be a sequence in independent, integrable, real-valued random variables. They are adapted to the filtration generated by the sequence, i.e. F$_n$ = σ(X$_0$, . . . , X$_n$) for all n ∈ $\mathbb {N} _{0}$. By (1) and (2), the Doob decomposition is given by
    $A_{n}=\sum _{k=1}^{n}{\bigl\mathbb {E} [X_{k}]-X_{k-1}{\bigr )},\quad n\in \mathbb {N} _{0},$
and
    $M_{n}=X_{0}+\sum _{k=1}^{n}{\biglX_{k}-\mathbb {E} [X_{k}]{\bigr )},\quad n\in \mathbb {N} _{0}.$
If the random variables of the original sequence X have mean zero, this simplifies to
    $A_{n}=-\sum _{k=0}^{n-1}X_{k}$ and $M_{n}=\sum _{k=0}^{n}X_{k},\quad n\in \mathbb {N} _{0},$
hence both processes are (possibly time-inhomogeneous) random walks. If the sequence X = (X$_n$)$_{n∈N0{\displaystyle\mathbb{N}_{0}}}$ 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 X = (X$_0$, X$_1$, . . . , X$_N$) denote the non-negative, discounted payoffs of an American option in a N-period financial market model, adapted to a filtration (F$_0$, F$_1$, . . . , F$_N$), and let $\mathbb {Q}$ denote an equivalent martingale measure. Let U = (U$_0$, U$_1$, . . . , U$_N$) 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 U = M + A denote the Doob decomposition with respect to $\mathbb {Q}$ of the Snell envelope U into a martingale M = (M$_0$, M$_1$, . . . , M$_N$) and a decreasing predictable process A = (A$_0$, A$_1$, . . . , A$_N$) with A$_0$ = 0. Then the largest stopping time to exercise the American option in an optimal way is
    $\tau _{\text{max}}:={\begin{cases}N&{\text{if }}A_{N}=0,\\\min\{n\in \{0,\dots ,N-1\}\mid A_{n+1}<0\}&{\text{if }}A_{N}<0.\end{cases}}$
Since A is predictable, the event {τ$_{max}$ = n} = {A$_n$ = 0, A$_{n+1}$ < 0} is in F$_n$ for every n ∈ {0, 1, . . . , N − 1}, hence τ$_{max}$ 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 τ$_{max}$ 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

* ↑ Doob (1953), see (Doob 1990, pp. 296−298)
* ↑ Durrett (2010)
* ↑ (Föllmer & Schied 2011, Proposition 6.1)
* ↑ (Williams 1991, Section 12.11, part (a) of the Theorem)
* ↑ (Williams 1991, Section 12.11, part (b) of the Theorem)
* ↑ (Lamberton & Lapeyre 2008, Chapter 2: Optimal stopping problem and American options)
* ↑ (Föllmer & Schied 2011, Chapter 6: American contingent claims)
* ↑ (Föllmer & Schied 2011, Proposition 6.10)
* ↑ (Föllmer & Schied 2011, Theorem 6.11)
* ↑ (Lamberton & Lapeyre 2008, Proposition 2.3.2)
* ↑ (Föllmer & Schied 2011, Theorem 6.21)
* ↑ (Schilling 2005, Problem 23.11)