📊 Stochastic Processes
Table of Contents
An Overview of Stochastic Processes
A stochastic process is a random process evolving with time. More precisely, a stochastic process is a collection of random variables indexed by by some mathematical set. If the stochastic process is characterized by a subset of nonnegative integers, it is called a discrete time (DT) stochastic process. If it is characterized by a subset of nonnegative real numbers, it is called a continuous time stochastic process. Discrete-time means equally-spaced points in time, separated by some time difference. In DT processes the behavior of a system is known only at discrete points in time and are not defined between those discrete points in time.
At each time, random variables will take values from a set, which is called the state space. The state space itself can be either discrete or continuous. Discrete state spaces are finite or countably infinite sets. Continuous state spaces are subsets of real numbers or n-dimensional space.
In contrast to the stochastic processes characterized by randomness involved in their evolution, there are other processes that evolve with time but are not stochastic. These counterparts of stochastic processes are called deterministic processes. The study of such processes leads to Differential equations (if time is continuous) and Difference equations (if time is discrete). Differential and Difference equations are important mainly in signal and system analysis because they describe the dynamic behavior of continuous-time and discrete-time physical systems respectively.