Definition 1.1: A Measurable space consists of a set and
a -algebra defined on .
Recall that a -algebra is simply a collection of subsets over that
is closed over unions, intersections and complements.
Definition 1.2: If and are -algebras
defined on and , then
is a sub -algebra of
An example: Say we have a probability triple defined over the experiment of tossing two independent, unbiased coins. Then,
The role of a -algebra in a probability triple is to define over what events probability is defined. If we toss two coins sequentially, our probability space would be , but after tossing the first coin, we wouldn’t be able to obtain the probability for events such as Both coins show the same face (as we simply don’t know the probability that the second coin turns up heads, or, in a more complicated setup, if the second coin’s toss is correlated with the first coin’s toss). An even simpler example is the state before we toss both coins: the coins are guaranteed to turn up some configuration in with probability 1, but we can’t pinpoint which configuration without knowing the probabilities associated with the coins.
-algebras, then, give us the ‘coarseness’ of the type of events that we’re allowed to play with. The trivial -algebra will give us either the event happening or not happening, and the complete -algebra will give us the probability of every possible type of event (provided we have a suitable measure over this space).
Filtrations
We’ll now explore the concept of a filtration: A filtration is formally defined as follows:
Definition 1.3: A filtration is a sequence of -algebras such that
Filtrations can be finite as well as infinite. A filtration basically captures the ‘coarseness’ of a set of events. Consider the same example as before. We’ll obtain three -algebras corresponding to the three states of the experiment1: no coins are tossed, one coin has been tossed and both coins have been tossed. The set of outcomes of a single coin is defined as and that for two coins is defined as . Then
Are the filtrations after these steps. Why only these? Looking at , we can obtain probabilities for the following events:
We can also write as the cross product of the -algebra of with :
This final argument lends itself to the generalization we seek: If we now have, say an infinite number of coins that we toss, then we can generate the sub -algebras
This is a filtration, as we have
What next?
This only scratches the tip of the surface on what filtrations are and what
they can do. Filtrations lend themselves well to stochastic processes
(basically sequences of random variables defined on the same probability
space). They are so common that we have the following definition:
Definition 1.4: A natural filtration of a stochastic process is defined
as (The -algebra generated by the random variables
upto ).
As an exercise, try to verify that this is indeed a filtration.
1 Why are there not three probability triples corresponding to the three states? A philosophical explanation is that they’re the same experiment, but a mathematical one is that even if we do have three triples, the sample space would be the same across all three, and all three probability measures would have to agree with each other. That is, for , we cannot have : the experiment remains the same (we’re tossing the same coins, so this is not possible). Also, note that , so if , then we’ll also have . Hence, it suffices to have just three -algebras