6-ConceptsProbability Page 181 Wednesday, February 4, 2004 3:00 PM
Concepts of Probability 181■ It is assumed that the economy is in one of the states of a probability
space (Ω,ℑ,P).■ Every security is described by two stochastic processes formed by two
time-dependent random variables St(ω) and dt(ω) representing prices
and cash flows of the asset.This representation is completely general and is not linked to the
assumption that the space of states is finite.INFORMATION STRUCTURES
Let’s now turn our attention to the question of time. The previous dis-
cussion considered a space formed by states in an abstract sense. We
must now introduce an appropriate representation of time as well as
rules that describe the evolution of information, that is, information
propagation, over time. The concepts of information and information
propagation are fundamental in economics and finance theory.
The concept of information in finance is different from both the
intuitive notion of information and that of information theory in which
information is a quantitative measure related to the a priori probability
of messages.^10 In our context, information means the (progressive) reve-
lation of the set of events to which the current state of the economy
belongs. Though somewhat technical, this concept of information sheds
light on the probabilistic structure of finance theory. The point is the
following. Assets are represented by stochastic processes, that is, time-
dependent random variables. But the probabilistic states on which these
random variables are defined represent entire histories of the economy.
To embed time into the probabilistic structure of states in a coherent
way calls for information structures and filtrations (a concept we
explain in the next section).
Recall that it is assumed that the economy is in one of many possible
states and that there is uncertainty on the state that has been realized.
Consider a time period of the economy. At the beginning of the period,
there is complete uncertainty on the state of the economy (i.e., there is
complete uncertainty on what path the economy will take). Different
events have different probabilities, but there is no certainty. As time
passes, uncertainty is reduced as the number of states to which the econ-(^10) There is indeed a deep link between information theory and econometrics embod-
ied in concepts such as the Fisher Information Matrix, see Chapter 12.