A stationary stochastic process is one whose probability distribution varies more or less constantly over a certain period of time.
In other words, a series of numbers may seem (and be) chaotic but take values within a limited range. Through this information models can be made that try to predict the variable. The daily returns of a financial asset are an example of stationary stochastic processes. Thus, the daily returns of the EURUSD, that is, the daily variation in percentage has the following form:
This chart reflects the daily returns in percent of the EURUSD since 1999. However, to better understand the concept, we will offer only the last 100 days.
By enlarging the graph we can see more clearly the behavior of the variable. During the last 100 days the EURUSD has had variations within the -1% and 1% range. We cannot predict what the variation will be on a specific day, but we can intuit (not confirm) the range of values within which the variable will be.
Are stationary stochastic processes predictable?
When referring to the predictability of a stationary stochastic process, you are not claiming that it is one hundred percent predictable. Reference is made to the possibility that with certain probability the series will take a range of values. An example is provided by the EURUSD daily returns chart. We cannot predict whether the EURUSD will rise or fall, but we can predict with a fairly high level of reliability that the EURUSD will have a return between -1 and 1%.
The following is a rough picture of the types of stochastic processes. Among which are stationary and non-stationary stochastic processes.