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.^[1]

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

Statement of the theorem

Let $(Ω, F, ℙ)$ be a probability space, $I = {0, 1, 2, . . . , N$ } with $N \in ℕ$ or $I = ℕ 0$ a finite or an infinite index set, $(F n) n \in I$ a filtration of F, and $X = (X n) n \in I$ an adapted stochastic process with $E[| X n |] < \infty$ for all $n \in I$ . Then there exists a martingale $M = (M n) n \in I$ and an integrable predictable process $A = (A n) n \in I$ starting with $A 0 = 0$ such that $X n = M n + A n$ for every $n \in I$ . Here predictable means that $A n$ is $F n -1$ -measurable for every $n ∈ I \ {0$ }. This decomposition is almost surely unique.^[2]^[3]^[4]

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.^[5] It is a supermartingale, if and only if $A$ is almost surely decreasing.

Remark

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

Proofs

Proof of the theorem

Existence

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

$A_n=\sum_{k=1}^n\bigl(\mathbb{E}[X_k\,|\,\mathcal{F}_{k-1}]-X_{k-1}\bigr)$

(1)

and

$M_n=X_0+\sum_{k=1}^n\bigl(X_k-\mathbb{E}[X_k\,|\,\mathcal{F}_{k-1}]\bigr),$

(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 $n + 1 \in I$ ) and $M n$ are $F n$ -measurable because the process $X$ is adapted, $E[| A n |] < \infty$ and $E[| M n |] < \infty$ because the process $X$ is integrable, and the decomposition $X n = M n + A n$ is valid for every $n \in I$ . 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 $n \in I$ , hence the decomposition is almost surely unique.

Proof of the corollary

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 (1) of $A$ is almost surely positive, hence $A$ is almost surely increasing. The equivalence for supermartingales is proved similarly.

Example

Let $X = (X n) n \inℕ 0$ 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 $n \in ℕ 0$ . 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}\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-inhomogenious) random walks. If the sequence $X = (X n) n \inℕ 0$ 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.^[6]^[7] 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 $ℚ$ denote an equivalent martingale measure. Let $U = (U 0, U 1, . . . , U N)$ denote the Snell envelope of $X$ with respect to $ℚ$ . The Snell envelope is the smallest $ℚ$ -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 $ℚ$ 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^[10]^[11] 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 $n \in {0, 1, . . . , N - 1$ }, 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 $ℚ$ .

Generalization

The Doob decomposition theorem can be generalized from probability spaces to σ-finite measure spaces.^[12]

Citations

↑ Doob (1953), see (Doob 1990, pp. 296−298)
↑ Durrett (2005)
↑ (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)

References

Doob, Joseph L. (1953), Stochastic Processes, New York: Wiley, ISBN 978-0-471-21813-5, MR 0058896, Zbl 0053.26802
Doob, Joseph L. (1990), Stochastic Processes (Wiley Classics Library ed.), New York: John Wiley & Sons, Inc., ISBN 0-471-52369-0, MR 1038526, Zbl 0696.60003
Durrett, Rick (2010), Probability: Theory and Examples, Cambridge Series in Statistical and Probabilistic Mathematics (4. ed.), Cambridge University Press, ISBN 978-0-521-76539-8, MR 2722836, Zbl 1202.60001
Föllmer, Hans; Schied, Alexander (2011), Stochastic Finance: An Introduction in Discrete Time, De Gruyter graduate (3. rev. and extend ed.), Berlin, New York: De Gruyter, ISBN 978-3-11-021804-6, MR 2779313, Zbl 1213.91006
Lamberton, Damien; Lapeyre, Bernard (2008), Introduction to Stochastic Calculus Applied to Finance, Chapman & Hall/CRC financial mathematics series (2. ed.), Boca Raton, FL: Chapman & Hall/CRC, ISBN 978-1-58488-626-6, MR 2362458, Zbl 1167.60001
Schilling, René L. (2005), Measures, Integrals and Martingales, Cambridge: Cambridge University Press, ISBN 978-0-52185-015-5, MR 2200059, Zbl 1084.28001
Williams, David (1991), Probability with Martingales, Cambridge University Press, ISBN 0-521-40605-6, MR 1155402, Zbl 0722.60001

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