2Martingales in discrete time
III Advanced Probability
2.1 Filtrations and martingales
We would like to model some random variable that “evolves with time”. For
example, in a simple random walk,
X
n
could be the position we are at time
n
.
To do so, we would like to have some
σ
-algebras
F
n
that tells us the “information
we have at time n”. This structure is known as a filtration.
Definition
(Filtration)
.
A filtration is a sequence of
σ
-algebras (
F
n
)
n≥0
such
that F ⊇ F
n+1
⊇ F
n
for all n. We define F
∞
= σ(F
0
, F
1
, . . .) ⊆ F.
We will from now on assume (Ω
, F, P
) is equipped with a filtration (
F
n
)
n≥0
.
Definition
(Stochastic process in discrete time)
.
A stochastic process (in discrete
time) is a sequence of random variables (X
n
)
n≥0
.
This is a very general definition, and in most cases, we would want
X
n
to
interact nicely with our filtration.
Definition (Natural filtration). The natural filtration of (X
n
)
n≥0
is given by
F
X
n
= σ(X
1
, . . . , X
n
).
Definition
(Adapted process)
.
We say that (
X
n
)
n≥0
is adapted (to (
F
n
)
n≥0
)
if X
n
is F
n
-measurable for all n ≥ 0. Equivalently, if F
X
n
⊆ F
n
.
Definition
(Integrable process)
.
A process (
X
n
)
n≥0
is integrable if
X
n
∈ L
1
for
all n ≥ 0.
We can now write down the definition of a martingale.
Definition
(Martingale)
.
An integrable adapted process (
X
n
)
n≥0
is a martingale
if for all n ≥ m, we have
E(X
n
| F
m
) = X
m
.
We say it is a super-martingale if
E(X
n
| F
m
) ≤ X
m
,
and a sub-martingale if
E(X
n
| F
m
) ≥ X
m
,
Note that it is enough to take
m
=
n −
1 for all
n ≥
0, using the tower
property.
The idea of a martingale is that we cannot predict whether
X
n
will go up
or go down in the future even if we have all the information up to the present.
For example, if
X
n
denotes the wealth of a gambler in a gambling game, then in
some sense (
X
n
)
n≥0
being a martingale means the game is “fair” (in the sense
of a fair dice).
Note that (
X
n
)
n≥0
is a super-martingale iff (
−X
n
)
n≥0
is a sub-martingale,
and if (
X
n
)
n≥0
is a martingale, then it is both a super-martingale and a sub-
martingale. Often, what these extra notions buy us is that we can formulate
our results for super-martingales (or sub-martingales), and then by applying the
result to both (
X
n
)
n≥0
and (
−X
n
)
n≥0
, we obtain the desired, stronger result
for martingales.