Doob decomposition theorem

In the theory of discrete-time stochastic processes, 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 starting at zero. The theorem was proved by and is named for Joseph Leo 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]

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 $A$ in 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 $X$ in 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 all $n ∈ I \ {0$ }. 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 submartingal, 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 $X = (X n) n \inℕ 0$ 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 last optimal exercise time of an American option.^[3]^[4] Let $X = (X 0, X 1, . . . , X T)$ denote the non-negative, discounted payoffs of an American option in a $T$ -period financial market model, adapted to a filtration $(F 0, F 1, . . . , F T)$ , and let $ℚ$ denote an equivalent martingale measure. Let $U = (U 0, U 1, . . . , U T)$ denote the Snell envelope of $X$ with respect to $ℚ$ . The Snell envelope is the smallest supermartingale dominating $X$ and represents a discounted, arbitrage-free price process of the American option. Let $U = M + A$ denote the Doob decomposition with respect to $ℚ$ of the Snell envelope $U$ into a martingale $M = (M 0, M 1, . . . , M T)$ and a decreasing predictable process $A = (A 0, A 1, . . . , A T)$ with $A 0 = 0$ . Then the last stopping time to exercise the American option in an optimal way is

\tau _{\text{max}}:={\begin{cases}T&{\text{if }}A_{T}=0,\\\min\{t\in \{0,\dots ,T-1\}\mid A_{t+1}<0\}&{\text{if }}A_{T}<0.\end{cases}}

Since $A$ is predictable, the event ${τ max = t} = {A t = 0, A t +1 < 0$ } is in $F t$ for every $t \in {0, 1, . . . , T - 1$ }, 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 $ℚ$ .

References

Doob, Joseph Leo (1953), Stochastic Processes, New York: Wiley, ISBN 978-0-471-21813-5, MR 0058896, Zbl 0053.26802 {{citation}}: ISBN / Date incompatibility (help)
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

[1] Doob (1953)

[2] Durrett (2005)

[3] Lamberton & Lapeyre (2008)

[4] Föllmer & Schied (2011)

[1]

[2]

[3]

[4]

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