In mathematicsspecifically, in the theory of stochastic processesDoob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American mathematician Joseph L. Doob.[1] Informally, the martingale convergence theorem typically refers to the result that any supermartingale satisfying a certain boundedness condition must converge. One may think of supermartingales as the random variable analogues of non-increasing sequences; from this perspective, the martingale convergence theorem is a random variable analogue of the monotone convergence theorem, which states that any bounded monotone sequence converges. There are symmetric results for submartingales, which are analogous to non-decreasing sequences.
A common formulation of the martingale convergence theorem for discrete-time martingales is the following. Let
X1,X2,X3,...
\supt
-] | |
\operatorname{E}[X | |
t |
<infty
where
- | |
X | |
t |
Xt
X
There is a symmetric statement for submartingales with bounded expectation of the positive part. A supermartingale is a stochastic analogue of a non-increasing sequence, and the condition of the theorem is analogous to the condition in the monotone convergence theorem that the sequence be bounded from below. The condition that the martingale is bounded is essential; for example, an unbiased
\pm1
As intuition, there are two reasons why a sequence may fail to converge. It may go off to infinity, or it may oscillate. The boundedness condition prevents the former from happening. The latter is impossible by a "gambling" argument. Specifically, consider a stock market game in which at time
t
Xt
The proof is simplified by making the (stronger) assumption that the supermartingale is uniformly bounded; that is, there is a constant
M
|Xn|\leqM
X1,X2,...
\liminfXn
\limsupXn
a
b
a<b
[a,b]
a
b
a
a
b
Consider a stock market game in which at time
t
Xt
N\inN
N
[a,b]
a
b
uN
N
N
(b-a)uN-2M
b-a
a\leqM
-M
0
\operatorname{E}[uN]\leq
2M | |
b-a |
.
By the monotone convergence theorem for expectations, this means that
\operatorname{E}[\limNuN]\leq
2M | |
b-a |
,
so the expected number of upcrossings in the whole sequence is finite. It follows that the infinite-crossing event for interval
[a,b]
0
a
b
1
a,b\inQ
[a,b]
1
Under the conditions of the martingale convergence theorem given above, it is not necessarily true that the supermartingale
(Xn)n
\limn\operatorname{E}[|Xn-X|]=0
As an example,[2] let
(Xn)n
\pm1
X0=1
N
Xn=0
(Yn)n
Yn:=Xmin(N,
N
(Xn)n
(Yn)n
(Yn)n
Y
Yn>0
Yn+1=Yn\pm1
Y
This means that
\operatorname{E}[Y]=0
\operatorname{E}[Yn]=1
n\geq1
(Yn)n
1
\operatorname{E}[Yn]=\operatorname{E}[Y0]=1
(Yn)n
(Yn)n
Y
(Yn)n
R
R
R=0
In the following,
(\Omega,F,F*,P)
F*=(Ft)t
N:[0,infty) x \Omega\toR
F*
0\leqs\leqt<+infty
Ns\geq\operatorname{E}[Nt\midFs].
Doob's first martingale convergence theorem provides a sufficient condition for the random variables
Nt
t\to+infty
\omega
\Omega
For
t\geq0
- | |
N | |
t |
=max(-Nt,0)
\supt\operatorname{E}[
- | |
N | |
t |
]<+infty.
Then the pointwise limit
N(\omega)=\limtNt(\omega)
exists and is finite for
P
\omega\in\Omega
It is important to note that the convergence in Doob's first martingale convergence theorem is pointwise, not uniform, and is unrelated to convergence in mean square, or indeed in any Lp space. In order to obtain convergence in L1 (i.e., convergence in mean), one requires uniform integrability of the random variables
Nt
The following are equivalent:
(Nt)t>0
\limC\supt
\int | |
\{\omega\in\Omega\mid|Nt(\omega)|>C\ |
N\inL1(\Omega,P;R)
Nt\toN
t\toinfty
P
L1(\Omega,P;R)
\operatorname{E}\left[\left|Nt-N\right|\right]=\int\Omega\left|Nt(\omega)-N(\omega)\right|dP(\omega)\to0ast\to+infty.
The following result, called Doob's upcrossing inequality or, sometimes, Doob's upcrossing lemma, is used in proving Doob's martingale convergence theorems.[3] A "gambling" argument shows that for uniformly bounded supermartingales, the number of upcrossings is bounded; the upcrossing lemma generalizes this argument to supermartingales with bounded expectation of their negative parts.
Let
N
(Xn)n
(l{F}n)n
a
b
a<b
(Un)n
Un
[n | |
i1 |
,
n | |
i2 |
]
n | |
i2 |
\leqn
X | |||||
|
<a<b<
X | |||||
|
[a,b]
(b-a)\operatorname{E}[Un]\le\operatorname{E}[(Xn-a)-].
X-
X
Let
M:[0,infty) x \Omega\toR
\supt\operatorname{E}[|Mt|p]<+infty
for some
p>1
M\inLp(\Omega,P;R)
Mt\toM
t\to+infty
P
Lp(\Omega,P;R)
The statement for discrete-time martingales is essentially identical, with the obvious difference that the continuity assumption is no longer necessary.
Doob's martingale convergence theorems imply that conditional expectations also have a convergence property.
Let
(\Omega,F,P)
X
L1
F*=(Fk)k
F
Finfty
(Fk)k
\operatorname{E}[X\midFk]\to\operatorname{E}[X\midFinfty]ask\toinfty
both
P
L1
This result is usually called Lévy's zero–one law or Levy's upwards theorem. The reason for the name is that if
A
Finfty
P[A\midFk]\to1A
P[A]=1A
P[A]\in\{0,1\}
Similarly we have the Levy's downwards theorem :
Let
(\Omega,F,P)
X
L1
(Fk)k
F
Finfty
\operatorname{E}[X\midFk]\to\operatorname{E}[X\midFinfty]ask\toinfty
both
P
L1
. Bernt Øksendal. Stochastic Differential Equations: An Introduction with Applications . Sixth. Springer. Berlin . 2003 . 3-540-04758-1. (See Appendix C)
. Rick Durrett. Probability: theory and examples. Second. Duxbury Press. 1996 . 978-0-534-24318-0.