Stochastic process

A stochastic process is a collection of random variables indexed by time, space, or another set. It models quantities whose values evolve or vary randomly.

Random variable: A measurable quantity whose value is determined by a chance experiment. Each indexed quantity in a stochastic process is a random variable.
Sample path: A function giving the values of a stochastic process across its index set for one outcome. It is the realized trajectory produced by the process.
Brownian motion: A continuous-time process with independent Gaussian increments and continuous sample paths. It models diffusion and underlies many continuous-time stochastic models.
Deterministic process: A process whose values are fixed by its initial conditions and governing rules, without randomness. It contrasts with models whose indexed values are random.
Probability distribution: A mathematical description of the possible values of a random variable and their probabilities. It specifies the possible value of a process at a chosen index.
Markov property: The condition that a system's future, given its present, is independent of its past. It simplifies process dynamics by limiting relevant history to the current state.
Poisson process: A counting process with independent, stationary increments and Poisson-distributed counts. It models randomly timed events such as arrivals.
Markov chain: A discrete-time stochastic process on a state space that satisfies the Markov property. It is the discrete-time, discrete-state Markov case.
Joint distribution: A probability distribution describing the values of two or more random variables together. It captures dependence between process values at different indices.
Transition probability: The conditional probability of moving from one state to another over a specified step or interval. Transition probabilities specify many discrete-state process dynamics.

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