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

Let ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}$ be a probability space, I = {0, 1, 2, ..., N} with ${\displaystyle N\in \mathbb {N} }$ or ${\displaystyle I=\mathbb {N} _{0}}$ a finite or countably infinite index set, ${\displaystyle ({\mathcal {F}}_{n})_{n\in I}}$ a filtration of ${\displaystyle {\mathcal {F}}}$, 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 A₀ = 0 such that X_n = M_n + A_n for every n ∈ I. Here predictable means that A_n is ${\displaystyle {\mathcal {F}}_{n-1}}$-measurable for every n ∈ I \ {0}. This decomposition is almost surely unique.^[2][3][4]

Remark

The theorem is valid word for word also for stochastic processes X taking values in the d-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{d}}$ or the complex vector space ${\displaystyle \mathbb {C} ^{d}}$. 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

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

(1)

and

${\displaystyle M_{n}=X_{0}+\sum _{k=1}^{n}{\bigl (}X_{k}-\mathbb {E} [X_{k}\,|\,{\mathcal {F}}_{k-1}]{\bigr )},}$

(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 ∈ 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

${\displaystyle \mathbb {E} [M_{n}-M_{n-1}\,|\,{\mathcal {F}}_{n-1}]=0}$    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

${\displaystyle \mathbb {E} [Y_{n}\,|\,{\mathcal {F}}_{n-1}]=Y_{n-1}}$    a.s.,

and also predictable, implying that

${\displaystyle \mathbb {E} [Y_{n}\,|\,{\mathcal {F}}_{n-1}]=Y_{n}}$    a.s.

for any n ∈ I \ {0}. Since Y₀ = A'₀ − A₀ = 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.^[5] It is a supermartingale, if and only if A is almost surely decreasing.

Proof

If X is a submartingale, then

${\displaystyle \mathbb {E} [X_{k}\,|\,{\mathcal {F}}_{k-1}]\geq X_{k-1}}$    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∈${\displaystyle \mathbb {N} _{0}}$} 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(X₀, . . . , X_n) for all n ∈ ${\displaystyle \mathbb {N} _{0}}$. By (1) and (2), the Doob decomposition is given by

${\displaystyle A_{n}=\sum _{k=1}^{n}{\bigl (}\mathbb {E} [X_{k}]-X_{k-1}{\bigr )},\quad n\in \mathbb {N} _{0},}$

and

${\displaystyle M_{n}=X_{0}+\sum _{k=1}^{n}{\bigl (}X_{k}-\mathbb {E} [X_{k}]{\bigr )},\quad n\in \mathbb {N} _{0}.}$

If the random variables of the original sequence X have mean zero, this simplifies to

${\displaystyle A_{n}=-\sum _{k=0}^{n-1}X_{k}}$    and    ${\displaystyle M_{n}=\sum _{k=0}^{n}X_{k},\quad n\in \mathbb {N} _{0},}$

hence both processes are (possibly time-inhomogeneous) random walks. If the sequence X = (X_n)_{n∈${\displaystyle \mathbb {N} _{0}}$} 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.^[6][7] Let X = (X₀, X₁, . . . , X_N) denote the non-negative, discounted payoffs of an American option in a N-period financial market model, adapted to a filtration (F₀, F₁, . . . , F_N), and let ${\displaystyle \mathbb {Q} }$ denote an equivalent martingale measure. Let U = (U₀, U₁, . . . , U_N) denote the Snell envelope of X with respect to ${\displaystyle \mathbb {Q} }$. The Snell envelope is the smallest ${\displaystyle \mathbb {Q} }$-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 ${\displaystyle \mathbb {Q} }$ of the Snell envelope U into a martingale M = (M₀, M₁, . . . , M_N) and a decreasing predictable process A = (A₀, A₁, . . . , A_N) with A₀ = 0. Then the largest stopping time to exercise the American option in an optimal way^[10][11] is

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

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 ${\displaystyle \mathbb {Q} }$.

Generalization

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

Citations

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