Stochastic Process

A stochastic process is a collection of random variables indexed by time or another parameter, representing the evolution of a random system.

Definition

A stochastic process is a family of random variables $\{X_t\}_{t \in T}$ defined on a probability_space $(\Omega, \mathcal{F}, P)$, where:

• $T$ is the index set (often representing time)
• Each $X_t : \Omega \to S$ maps outcomes to a state space $S$

Classification

By Index Set $T$

• Discrete-time: $T = \mathbb{N}$ or $T = \mathbb{Z}$ (e.g., random walks)
• Continuous-time: $T = [0, \infty)$ or $T = \mathbb{R}$ (e.g., Brownian motion)

By State Space $S$

• Discrete state space: $S$ is countable (e.g., Markov chains)
• Continuous state space: $S = \mathbb{R}$ or $\mathbb{R}^n$ (e.g., diffusion processes)

Examples

• Random walks
• Markov chains
• Brownian motion (Wiener process)
• Poisson processes
• Martingales

Related

• probability_space
• probability_measure