6-ConceptsProbability Page 183 Wednesday, February 4, 2004 3:00 PM
Concepts of Probability 183tion of information is called a filtration. In the discrete case, however, the
two concepts—information structure and filtration—are equivalent.
The concept of filtration is based on identifying all events that are
known at any given instant. It is assumed that it is possible to associate
to each trading moment t a σ-algebra of events ℑt ⊂ ℑ formed by all
events that are known prior to or at time t. It is assumed that events are
never “forgotten,” that is, that ℑt ⊂ ℑs, if t < s. An ordering of time is
thus created. This ordering is formed by an increasing sequence of σ-
algebras, each associated to the time at which all its events are known.
This sequence is a filtration. Indicated as {ℑt}, a filtration is therefore an
increasing sequence of all σ-algebras ℑt, each associated to an instant t.
In the finite case, it is possible to create a mutual correspondence
between filtrations and information structures. In fact, given an infor-
mation structure, it is possible to associate to each partition the algebra
generated by the same partition. Observe that a tree information struc-
ture is formed by partitions that create increasing refinement: By going
from one instant to the next, every set of the partition is decomposed.
One can then conclude that the algebras generated by an information
structure form a filtration.
On the other hand, given a filtration {ℑt}, it is possible to associate a
partition to each ℑt. In fact, given any element that belongs to Ω, con-
sider any other element that belongs to Ω such that, for each set of ℑt,
both either belong to or are outside this set. It is easy to see that classes
of equivalence are thus formed, that these create a partition, and that
the algebra generated by each such partition is precisely the ℑt that has
generated the partition.
A stochastic process is said to be adapted to the filtration {ℑt} if the
variable Xt is measurable with respect to the σ-algebra ℑt. It is assumed
that the price and cash distribution processes St(ω) and dt(ω) of every
asset are adapted to {ℑt}. This means that, for each t, no measurement
of any price or cash distribution variable can identify events not
included in the respective algebra or σ-algebra. Every random variable
is a partial image of the set of states seen from a given point of view and
at a given moment.
The concepts of filtration and of processes adapted to a filtration
are fundamental. They ensure that information is revealed without
anticipation. Consider the economy and associate at every instant a par-
tition and an algebra generated by the partition. Every random variable
defined at that moment assumes a value constant on each set of the par-
tition. The knowledge of the realized values of the random variables
does not allow identifying sets of events finer than partitions.
One might well ask: Why introduce the complex structure of σ-alge-
bras as opposed to simply defining random variables? The point is that,