1.7. STOCHASTIC PROCESSES 63
of elementary probability theory. In elementary probability theory, random
variables are a mapping from a sample space to the real numbers, for stochas-
tic processes the mapping is from a sample space to a space of functions. Now
we can ask questions like
- “What is the probability of the set of functions that exceed a fixed
value on a fixed time interval?” - “What is the probability of the set of functions having a certain limit
at infinity?” - “What is the probability of the set of functions which are differentiable
everywhere?”
This is a fruitful way to consider stochastic processes, but it requires sophis-
ticated mathematical tools and careful analysis.
Another way to look at stochastic processes is to ask what happens at
special times. For example, one can consider the time it takes until the
function takes on one of two certain values, sayaandbto be specific. Then
one can ask “What is the probability that the stochastic process assumes
the valuea before it assumes the valueb?” Note that here the time that
each function assumes the valueais different, it becomes a random time.
This provides an interaction between the time variable and the sample point
through the values of the function. This too is a fruitful way to think about
stochastic processes.
In this text, we will consider each of these approaches with the corre-
sponding questions.
Sources
The material in this section is adapted from many texts on probability theory
and stochastic processes, especially the classic texts by S. Karlin and H.
Taylor, S. Ross, and W. Feller.
Problems to Work for Understanding
Outside Readings and Links:
- Origlio, Vincenzo. “Stochastic.” From MathWorld–A Wolfram Web
Resource, created by Eric W. Weisstein. Stochastic