Stochastic Series - Pt 0 - What is a stochastic process?
This is quite a hard question to answer straight away, so perhaps we should start with a different question.
What is a deterministic process?
One example is the weekdays, we know that Tuesday comes after Monday. Another example is a traffic light following a fixed cycle, where red always comes immediately after yellow:
P(RED | YELLOW) = 1
The probability of transitioning into red from yellow is 1 or 100%, there is no randomness at all here.
Technically, a deterministic process is a special case of a stochastic process, but the distinction is useful when talking about randomness.
However, some transitions can be certain even within a stochastic process. Let's say we use the number of customers in a shop as the state and observe it at hourly steps.
Let Xₜ be the number of customers in the shop at time t.
P(X₈ₚₘ > 100) = 0.2
This reads as: The probability of more than 100 customers being in the shop at 8 pm is equal to 0.2 or 20%.
But if we know that the store closes at 9 pm and all customers have left by 10 pm, we can be certain that:
P(X₁₀ₚₘ = 0) = 1
The probability of zero customers being in the shop at 10 pm is equal to 1 or 100%.
So the state at one time can be certain within a stochastic process even though the states at other times are uncertain.
Randomness
A stochastic process can include randomness, where even if we know the current state, we are not sure what state we will transition into.
The next state might be independent of the previous state, like repeatedly throwing a fair die:
P(6 | 3) = 1/6
The probability of throwing a six after throwing a three is always 1/6, the previous state (three) does not tell us anything about the next state. Each throw is a random event, and the sequence of outcomes forms a stochastic process.
The probabilities might also depend on context, like traffic:
P(TRAFFIC JAM | MONDAY, 08:00) = 0.6
In this example, the probability of a traffic jam GIVEN that it is Monday at 08:00 is 0.6 or 60%. Observing the traffic state over time gives us a stochastic process.
So a stochastic process can be defined as a mathematical model describing a quantity whose value can vary randomly over time.