Stochastic Process → Definition

Before saying anything about stochastic process, we need to recall what deterministic system (or deterministic process) is?

Deterministic System
A system in which the later states of the system determined by the earlier states is called deterministic system.

For example, if a company defaults or face heavy loss, it is determined that the shareholders (who owns the company) will also have to face the loss.

Stochastic Process
The stochastic process (also called random process) is not wholly determined by the earlier one stage, but, in this case, we take observations of many random values over a period of time. So, stochastic process is a collection of random variables. And, it is a probabilistic process.

For example, we are interested in stock price of a particular company. And, we observe the stock prices at different time periods.

S(0) is the stock price at time = 0. S(1) is the stock prices at time = 1, and so on. These stock prices can take any value, so , are random variables. And, that the stock prices can move up or down, so they are independent.

Definition of Stochastic Process
Let N(t₁) be a random value when t=1, and N(t₂) be another random value when t=2, and so on.. And, when we consider N(t) for t∈[0,∞] collectively. We say N(t) is a stochastic process or random process.

One of the most widely used stochastic process is Poisson Process.