Doob decomposition theorem

Template:Short description 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 $(\mathrm{\Omega },ℱ,\mathbb{P})$ be a probability space, Template:Math with $N\in \mathbb{N}$ or $I={\mathbb{N}}_0$ a finite or countably infinite index set, $({ℱ}_n{)}_{n\in I}$ a filtration of $ℱ$, and Template:Math an adapted stochastic process with Template:Math for all Template:Math. Then there exist a martingale Template:Math and an integrable predictable process Template:Math starting with Template:Math such that Template:Math for every Template:Math. Here predictable means that Template:Math is ${ℱ}_{n-1}$-measurable for every Template:Math. This decomposition is almost surely unique.

Remark

The theorem is valid word for word also for stochastic processes Template:Math taking values in the Template:Math-dimensional Euclidean space ${ℝ}^d$ or the complex vector space ${ℂ}^d$. This follows from the one-dimensional version by considering the components individually.

Proof

Existence

Using conditional expectations, define the processes Template:Math and Template:Math, for every Template:Math, explicitly by

and

where the sums for Template:Math are empty and defined as zero. Here Template:Math adds up the expected increments of Template:Math, and Template:Math adds up the surprises, i.e., the part of every Template:Math that is not known one time step before. Due to these definitions, Template:Math (if Template:Math) and Template:Math are Template:Math-measurable because the process Template:Math is adapted, Template:Math and Template:Math because the process Template:Math is integrable, and the decomposition Template:Math is valid for every Template:Math. The martingale property

$𝔼[M_n-M_{n-1}|{ℱ}_{n-1}]=0$    a.s.

also follows from the above definition (Template:EquationNote), for every Template:Math}.

Uniqueness

To prove uniqueness, let Template:Math be an additional decomposition. Then the process Template:Math is a martingale, implying that

$𝔼[Y_n|{ℱ}_{n-1}]=Y_{n-1}$    a.s.,

and also predictable, implying that

$𝔼[Y_n|{ℱ}_{n-1}]=Y_n$    a.s.

for any Template:Math}. Since Template:Math by the convention about the starting point of the predictable processes, this implies iteratively that Template:Math almost surely for all Template:Math, hence the decomposition is almost surely unique.

Corollary

A real-valued stochastic process Template:Math is a submartingale if and only if it has a Doob decomposition into a martingale Template:Math and an integrable predictable process Template:Math that is almost surely increasing. It is a supermartingale, if and only if Template:Math is almost surely decreasing.

Proof

If Template:Math is a submartingale, then

$𝔼[X_k|{ℱ}_{k-1}]\ge X_{k-1}$    a.s.

for all Template:Math}, which is equivalent to saying that every term in definition (Template:EquationNote) of Template:Math is almost surely positive, hence Template:Math is almost surely increasing. The equivalence for supermartingales is proved similarly.

Example

Let Template:Math be a sequence in independent, integrable, real-valued random variables. They are adapted to the filtration generated by the sequence, i.e. Template:Math for all $n\in {\mathbb{N}}_0.$ By (Template:EquationNote) and (Template:EquationNote), the Doob decomposition is given by

$A_n=\sum _{k=1}^n(𝔼[X_k]-X_{k-1}),n\in {\mathbb{N}}_0,$

and

$M_n=X_0+\sum _{k=1}^n(X_k-𝔼[X_k]),n\in {\mathbb{N}}_0.$

If the random variables of the original sequence Template:Math have mean zero, this simplifies to

$A_n=-\sum _{k=0}^{n-1}{X}_k$    and    $M_n=\sum _{k=0}^n{X}_k,n\in {\mathbb{N}}_0,$

hence both processes are (possibly time-inhomogeneous) random walks. If the sequence Template:Math consists of symmetric random variables taking the values Template:Math and Template:Math, then Template:Math is bounded, but the martingale Template:Math and the predictable process Template:Math are unbounded simple random walks (and not uniformly integrable), and Doob's optional stopping theorem might not be applicable to the martingale Template:Math 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 Template:Math denote the non-negative, discounted payoffs of an American option in a Template:Math-period financial market model, adapted to a filtration Template:Math, and let $\mathbb{Q}$g denote an equivalent martingale measure. Let Template:Math denote the Snell envelope of Template:Math with respect to $\mathbb{Q}$. The Snell envelope is the smallest $\mathbb{Q}$-supermartingale dominating Template:Math and in a complete financial market it represents the minimal amount of capital necessary to hedge the American option up to maturity. Let Template:Math denote the Doob decomposition with respect to $\mathbb{Q}$ of the Snell envelope Template:Math into a martingale Template:Math and a decreasing predictable process Template:Math with Template:Math. Then the largest stopping time to exercise the American option in an optimal way is

${\tau }_{\text{max}}:=\{\begin{array}{cc}N& \text{if }{A}_N=0,\\ min\{n\in \{0,\dots ,N-1\}\mid A_{n+1}<0\}& \text{if }{A}_N<0.\end{array}$

Since Template:Math is predictable, the event Template:Math} is in Template:Math for every Template:Math}, hence Template:Math 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 Template:Math the discounted value process Template:Math is a martingale with respect to $\mathbb{Q}$.

Generalization

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

References

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