Stochastic Modelling

In many real-world systems, randomness does not occur in isolation. Instead, it unfolds over time, with the current state of a system often depending on its past behaviour. Examples include the number of customers in a queue throughout the day, daily stock prices, rainfall levels over a year, or the spread of an infectious disease. To model such evolving random phenomena, we require a more general framework than single random variables. This leads us to stochastic processes.

A stochastic process is a collection of random variables indexed by time (or space), typically written as

\[ \{X_t : t \in T\}, \]

where each \(X_t\) represents the state of the system at time \(t\). Rather than studying one outcome, we now study an entire trajectory (or path) of outcomes across time. This allows us to capture not just variability, but also dynamics, dependence, and temporal structure.

This chapter builds on earlier concepts in simulation, where randomness, repetition, and modelling are combined to study complex systems. Previously, we focused on generating random variables from known distributions. Now, we extend this idea to generating sequences of dependent random variables, which is essential for modelling systems such as:

• Queueing systems (customers arriving and being served over time)
• Epidemic models (number of infected individuals evolving)
• Financial time series (asset prices changing daily)
• Environmental processes (temperature, pollution, rainfall)