STOCHASTIC PROCESSES 01

What Is a Stochastic Process?

A stochastic process is a collection of random variables describing how an uncertain system evolves through time.

The main idea

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

For each time \(t\), \(X_t\) is a random variable. Together, these random variables describe the random evolution of the system.

1. From one random variable to a process

A random variable describes one uncertain quantity. A stochastic process describes an uncertain quantity at many times.

\[X_0,\;X_1,\;X_2,\ldots\]

or, in continuous time,

\[\{X_t:t\ge0\}.\]

2. Example

Let \(I_t\) be the number of infectious individuals at time \(t\). Because infection and recovery events are uncertain, \(I_t\) is random. The collection \(\{I_t:t\ge0\}\) is a stochastic process.

3. Index set

The set \(T\) tells us when the process is observed. It may be discrete, such as \(T=\{0,1,2,\ldots\}\), or continuous, such as \(T=[0,\infty)\).

4. State space

The state space contains all values the process can take. For an epidemic count it might be \(\{0,1,\ldots,N\}\).

5. One process, many possible paths

The stochastic model specifies a probability law for many possible evolutions. One realised evolution is called a trajectory or sample path.

\[\boxed{\text{probability law}\rightarrow\text{many possible trajectories}}\]

6. Deterministic comparison

A deterministic model gives one path once the initial condition is fixed. A stochastic model gives a distribution over possible paths.

7. Why stochastic processes matter

They model systems in which uncertainty changes through time: epidemics, populations, queues, financial prices, molecular systems and noisy physical motion.