properties of the flow. This is analogus to using linear models in the 
deterministic case. If the predictions agree with the experimental 
results, the linear theory is said to be good; if they do not, then the 
process is said to be nonlinear. In using a statistical model, one 
should recognize that it is only a model and not the actual process, and 
one should continually strive to determine the accuracy of his models. 
There are many reasons in favor of applying stochastic theory. The 
solution of the stochastic problem may be possible whereas the determin- 
istic theory may be hopelessly impossible. In many problems such as 
that of traffic flow, one may not be interested in the microscopic 
properties but merely in certain macroscopic properties. In the control 
problem, the stochastic model distinguishes between open and closed 
looped systems but the deterministic model does not. Another reason for 
using a stochastic model may be that this model is closer to the physics 
of the actual situation. 
In any case the purpose of this section is to lay the ground work 
for stochastic control theory. Our attention will be focused on certain 
concepts of stochastic processes and random differential equations. 
To describe a stochastic process rigorously would require measure 
theory and a great deal more time. Our approach will therefore not be 
rigorous, but hopefully it will be complete enough to get across the 
basic ideas. For the rigorous approach, see either Doon or Gikhman and 
Skoralnodses 
A real random variable € is a set of numbers or events together 
with a probability measure defined on this set. It is characterized by 
its distribution function F(x) which is defined by 
Ge) = Pte < ox} 
“Nee. J. L., "Stochastic Processes,'' Wiley, Inc., New York (1963). 
MU ecikdbaneermy, I. I. and A. V. Skorokhod, "Introduction to the Theory of 
Random Processes,'' W. B. Saunders Company, Philadelphia, Pa. (1969). 
42 
