- Source: 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.
The analogous theorem in the continuous-time case is the Doob–Meyer decomposition theorem.
Statement
Let
(
Ω
,
F
,
P
)
{\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}
be a probability space, I = {0, 1, 2, ..., N} with
N
∈
N
{\displaystyle N\in \mathbb {N} }
or
I
=
N
0
{\displaystyle I=\mathbb {N} _{0}}
a finite or countably infinite index set,
(
F
n
)
n
∈
I
{\displaystyle ({\mathcal {F}}_{n})_{n\in I}}
a filtration of
F
{\displaystyle {\mathcal {F}}}
, 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
F
n
−
1
{\displaystyle {\mathcal {F}}_{n-1}}
-measurable for every n ∈ I \ {0}.
This decomposition is almost surely unique.
= Remark
=The theorem is valid word for word also for stochastic processes X taking values in the d-dimensional Euclidean space
R
d
{\displaystyle \mathbb {R} ^{d}}
or the complex vector space
C
d
{\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
and
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
E
[
M
n
−
M
n
−
1
|
F
n
−
1
]
=
0
{\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
E
[
Y
n
|
F
n
−
1
]
=
Y
n
−
1
{\displaystyle \mathbb {E} [Y_{n}\,|\,{\mathcal {F}}_{n-1}]=Y_{n-1}}
a.s.,
and also predictable, implying that
E
[
Y
n
|
F
n
−
1
]
=
Y
n
{\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 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. It is a supermartingale, if and only if A is almost surely decreasing.
= Proof
=If X is a submartingale, then
E
[
X
k
|
F
k
−
1
]
≥
X
k
−
1
{\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 = (Xn)n∈
N
0
{\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. Fn = σ(X0, . . . , Xn) for all n ∈
N
0
{\displaystyle \mathbb {N} _{0}}
. By (1) and (2), the Doob decomposition is given by
A
n
=
∑
k
=
1
n
(
E
[
X
k
]
−
X
k
−
1
)
,
n
∈
N
0
,
{\displaystyle 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
+
∑
k
=
1
n
(
X
k
−
E
[
X
k
]
)
,
n
∈
N
0
.
{\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
A
n
=
−
∑
k
=
0
n
−
1
X
k
{\displaystyle A_{n}=-\sum _{k=0}^{n-1}X_{k}}
and
M
n
=
∑
k
=
0
n
X
k
,
n
∈
N
0
,
{\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 = (Xn)n∈
N
0
{\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. 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
Q
{\displaystyle \mathbb {Q} }
denote an equivalent martingale measure. Let U = (U0, U1, . . . , UN) denote the Snell envelope of X with respect to
Q
{\displaystyle \mathbb {Q} }
. The Snell envelope is the smallest
Q
{\displaystyle \mathbb {Q} }
-supermartingale dominating X and in a complete financial market it represents the minimal amount of capital necessary to hedge the American option up to maturity. Let U = M + A denote the Doob decomposition with respect to
Q
{\displaystyle \mathbb {Q} }
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 is
τ
max
:=
{
N
if
A
N
=
0
,
min
{
n
∈
{
0
,
…
,
N
−
1
}
∣
A
n
+
1
<
0
}
if
A
N
<
0.
{\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 {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
Q
{\displaystyle \mathbb {Q} }
.
Generalization
The Doob decomposition theorem can be generalized from probability spaces to σ-finite measure spaces.
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
Kata Kunci Pencarian:
- Doob decomposition theorem
- Doob–Meyer decomposition theorem
- Joseph L. Doob
- Decomposition (disambiguation)
- Kramkov's optional decomposition theorem
- List of theorems
- Central limit theorem
- Gaussian random field
- List of statistics articles
- Autoregressive model
