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 be a probability space, I = {0, 1, 2, ..., N} with or a finite or countably infinite index set, a filtration of , and X = (Xn)n∈I an adapted stochastic process with E[|Xn|] < ∞ for all n ∈ I. Then there exist a martingale M = (Mn)n∈I and an integrable predictable process A = (An)n∈I starting with A0 = 0 such that Xn = Mn + An for every n ∈ I. Here predictable means that An is -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 or the complex vector space . 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
(1) |
and
(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 Xk that is not known one time step before. Due to these definitions, An+1 (if n + 1 ∈ I) and Mn are Fn-measurable because the process X is adapted, E[|An|] < ∞ and E[|Mn|] < ∞ because the process X is integrable, and the decomposition Xn = Mn + An is valid for every n ∈ I. The martingale property
- 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
- a.s.,
and also predictable, implying that
- 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 Yn = 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
- 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 = (Xn)n∈ be a sequence in independent, integrable, real-valued random variables. They are adapted to the filtration generated by the sequence, i.e. Fn = σ(X0, . . . , Xn) for all n ∈ . By (1) and (2), the Doob decomposition is given by
and
If the random variables of the original sequence X have mean zero, this simplifies to
- and
hence both processes are (possibly time-inhomogeneous) random walks. If the sequence X = (Xn)n∈ 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, . . . , XN) denote the non-negative, discounted payoffs of an American option in a N-period financial market model, adapted to a filtration (F0, F1, . . . , FN), and let denote an equivalent martingale measure. Let U = (U0, U1, . . . , UN) denote the Snell envelope of X with respect to . The Snell envelope is the smallest -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 of the Snell envelope U into a martingale M = (M0, M1, . . . , MN) and a decreasing predictable process A = (A0, A1, . . . , AN) with A0 = 0. Then the largest stopping time to exercise the American option in an optimal way[10][11] is
Since A is predictable, the event {τmax = n} = {An = 0, An+1 < 0} is in Fn for every n ∈ {0, 1, . . . , N − 1}, hence τmax 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 τmax the discounted value process U is a martingale with respect to .
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 (2010)
- ^ (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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