What is Stochastic Process?

Random process or stochastic process is a family of model on dynamic random states. Dynamic means time dependent. States are the internal properties of a system that together with the inputs to the system will reveal the output of a system. The states can be discrete or continuous. The time domain can also be discrete or continuous. Thus, stochastic process describes time-dependent states of a system. The states are represented by random variable \( X_t \). Actually \( X_t \) is a random function that represents a trajectory of the states over time. Each realization \( x_k \) of a random variable \( X_t \) is a random function that produces different results (as shown in left figure below). The set of all possible realization forms the functional space of the random function. The middle and right figures below show respectively the linear and geometric random function and their boundaries with a sample of one realization.

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Preferable reference for this tutorial is

Teknomo, Kardi. (2019) Stochastic Process Tutorial .
http://people.revoledu.com/kardi/tutorial/StochasticProcess/