Stochastic Process – Definition, Classification, Types and Facts

Stochastic Process Definition

Stochastic Process – Definition: A stochastic process is a mathematical model that describes a sequence of random variables. The individual random variables in the sequence usually called “events.”

A stochastic process can be used to model the evolution of a physical system over time, or the evolution of a random variable over time.

The most common type of stochastic process is a Markov process.

Types of Stochastic Processes

There are four types of stochastic processes:

1. Discrete-time stochastic processes: These processes are characterized by a sequence of random variables, each of which takes on a finite set of values.
2. Continuous-time stochastic processes: These processes are characterized by a sequence of random variables, each of which takes on a continuous range of values.
3. Stationary stochastic processes: These processes are characterized by the fact that the statistical properties of the random variables do not change over time.
4. Non-stationary stochastic processes: These processes are characterized by the fact that the statistical properties of the random variables change over time.

Classification of Stochastic processes

A stochastic process can classified by its type of stochasticity, or the manner in which the random variables are generated. The five most common types of stochasticity are:

1. Discrete-time stochastic processes: In discrete-time stochastic processes, the random variables are generated by taking a finite number of values over a fixed time interval.
2. Continuous-time stochastic processes: In continuous-time stochastic processes, the random variables are generated by taking an infinite number of values over an infinite time interval.
3. Stationary stochastic processes: A stationary stochastic process is one in which the statistical properties of the random variables do not change over time.
4. Ergodic stochastic processes: An ergodic stochastic process is one in which the statistical properties of the random variables do change over time, but the process eventually settles down to a stationary state.
5. Non-ergodic stochastic processes: A non-ergodic stochastic process is one in which the statistical properties of the random variables never settle down to a stationary state.