Stochastic process

Summary

A stochastic process is a family $(X_t)_{t \in T}$ of random variables on one probability space $(\Omega, \mathcal{F}, P)$, all with values in the same measurable space $(S, \mathcal{S})$. The set $T$ is the index set, often a set of times such as $\{0, 1, 2, \dots\}$ or $[0, \infty)$. The set $S$ is the state space. For each outcome $\omega \in \Omega$, the function $t \mapsto X_t(\omega)$ is a sample path of the process.

Remarks

The process is in discrete time if $T$ is countable, such as $\{0, 1, 2, \dots\}$, and in continuous time if $T$ is an interval. The finite-dimensional distributions are the laws of the vectors $(X_{t_1}, \dots, X_{t_n})$ for finite sets $\{t_1, \dots, t_n\} \subseteq T$. The Kolmogorov extension theorem builds a process from a consistent family of such laws. Important examples are the Bernoulli process, random walks, Markov chains, the Poisson process and the Wiener process.

Explanations

L1 human

A stochastic process is a random quantity that changes with time. Examples are the price of a share day by day and the number of customers in a queue. At each time it is a random variable, and one run of the whole process gives a random path.