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 $$N \in \N$$ or $$I = \N_0$$ 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 $$\Reals^d$$ or the complex vector space $$\Complex^d$$. This follows from the one-dimensional version by considering the components individually.

Existence
Using conditional expectations, define the processes $A$ and $M$, for every $n ∈ I$, explicitly by

and

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 $\mathcal{F}_{n}$-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 ($$), 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


 * $$\mathbb{E}[X_k\,|\,\mathcal{F}_{k-1}]\ge X_{k-1}$$   a.s.

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 $X = (X_{n})_{n∈$\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. $\mathcal{F}_{n} = σ(X_{0}, . . ., X_{n})$ for all $n ∈ $\mathbb{N}_0$$. By ($$) and ($$), 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


 * $$M_n=X_0+\sum_{k=1}^{n}\bigl(X_k-\mathbb{E}[X_k]\bigr),\quad n\in\mathbb{N}_0.$$

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 $X = (X_{n})_{n∈$\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 $(\mathcal{F}_{0}, \mathcal{F}_{1}, . . ., \mathcal{F}_{N})$, 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 $\mathcal{F}_{n}$ 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.