1.7. STOCHASTIC PROCESSES 61
and the collisions of molecules in a gas are a “random walk” responsible for
diffusion. In this process, we measure the position of the particle over time
so that is a stochastic process from the non-negative real numbers to either
one-, two- or three-dimensional real space. Random walks have fascinating
mathematical properties. Scientists can make the model more realistic by
including the effects of inertia leading to a more refined form of Brownian
motion called the “Ornstein-Uhlenbeck process”.
Extending this idea to economics, we will model market prices of financial
assets such as stocks as a continuous time, continuous space process. Ran-
dom market forces create small but constantly occurring price changes. This
results in a stochastic process from a continuous time variable representing
time to the reals or non-negative reals representing prices. By refining the
model so that prices can only be non-negative leads to the stochastic process
known as “geometric Brownian motion”.
1.7 The family tree of some stochastic processes
A sequence or interval of random outcomes, that is to say, a string of random
outcomes dependent on time as well as the randomness is called astochas-
tic process. Because of the inclusion of a time variable, the rich range of
random outcome distributions is multiplied to an almost bewildering variety
of stochastic processes. Nevertheless, the most commonly studied types of
random processes do have a family tree of relationships. My interpretation
of the family tree is included below, along with an indication of the types
studied in this course.
Ways to Interpret Stochastic Processes
Stochastic processes are functions of two variables, the time index and the
sample point. As a consequence, there are several ways to represent the
stochastic process. The simplest is to look at the stochastic process at a fixed
value of time. The result is a random variable with a probability distribution,
just as studied in elementary probability.
Another way to look at a stochastic process is to consider the stochastic
process as a function of the sample pointω. For eachω there is an associ-
ated functionX(t). This means that one can look at a stochastic process as
a mapping from the sample space Ω to a set of functions. In this interpre-
tation, stochastic processes are a generalization from the random variables