Wiener process

A continuous-time stochastic process with independent, normally distributed increments and continuous paths. It is the standard mathematical model of Brownian motion.

Stochastic process: A collection of random variables indexed by time or another parameter. A Wiener process is a specific continuous-time member of this broader class.
Brownian scaling: The invariance of Brownian motion under scaling time by a factor and space by its square root. This scaling follows directly from the increment variances and identifies the process's characteristic time-space balance.
Brownian motion: The irregular motion of particles suspended in a fluid, also modeled mathematically by a diffusion process. The Wiener process formalizes the random motion first observed in suspended particles.
Stochastic differential equation: A differential equation whose dynamics include random forcing, often represented by Brownian motion. Wiener processes supply the noise term in many continuous-time stochastic models.
Lévy process: A stochastic process with stationary independent increments and paths that are right-continuous with left limits. Wiener motion is the continuous-path Gaussian case within this broader family.
Normal distribution: A symmetric probability distribution characterized by its mean and variance. Every Wiener-process increment has a normal distribution whose variance equals the elapsed time.
Markov property: A property stating that the future depends on the past only through the present state. Independent increments give the Wiener process this memoryless conditional behavior.
Robert Brown: A Scottish botanist who described the irregular motion of microscopic particles suspended in fluid in 1827. His observations gave the physical phenomenon later modeled by the Wiener process its name.
Black–Scholes model: A mathematical model for pricing options under assumptions about asset-price dynamics and markets. Its standard derivation models asset-price fluctuations using a Wiener process.
Poisson process: A counting process with independent increments that records events occurring at a constant average rate. Unlike Wiener motion, it has discrete jumps and nonnegative integer-valued states.

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