Brownian motion

The term Brownian motion (in honor of the botanist Robert Brown) refers to either

Description of the mathematical model

Mathematically, Brownian motion is a Wiener process in which the conditional probability distribution of the particle's position at time t + dt, given that its position at time t is p, is a normal distribution with a mean of p + μ dt and a variance of σ2 dt; the parameter μ is the drift velocity, and the parameter σ2 is the power of the noise. These properties clearly establish that Brownian motion is Markovian (i.e. it satisfies the Markov property). Brownian motion is related to the random walk problem and it is generic in the sense that many different stochastic processes reduce to Brownian motion in suitable limits.

Related Topics:
Wiener process - Normal distribution - Mean - Variance - Markov property - Random walk

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In fact, the Wiener process is the only time-homogeneous stochastic process with independent increments that has continuous trajectories. These are all reasonable approximations to the physical properties of Brownian motion.

Related Topics:
Homogeneous - Stochastic process - Independent increments

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The mathematical theory of Brownian motion has been applied in contexts ranging far beyond the movement of particles in fluids. For example, in the modern theory of option pricing, asset classes are sometimes modeled as if they move according to a closely related process, geometric Brownian motion.

Related Topics:
Option pricing - Geometric Brownian motion

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It turns out that the Wiener process is not a physically realistic model of the motion of Brownian particles. More sophisticated formulations of the problem have led to the mathematical theory of diffusion processes. The accompanying equation of motion is called the Langevin equation or the Fokker-Planck equation depending on whether it is formulated in terms of random trajectories or probability densities.

Related Topics:
Diffusion processes - Langevin equation - Fokker-Planck equation

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