Brownian motion


One-dimensional Brownian motionMathworldPlanetmath is a stochastic processMathworldPlanetmath W(t), defined for t[0,) such that

  1. 1.

    W(0)=0 almost surely

  2. 2.

    The sample paths tW(t) are almost surely continuous.

  3. 3.

    For any finite sequencePlanetmathPlanetmath of times t0<t1<<tn, the increments


    are independent.

  4. 4.

    For any times s<t, W(t)-W(s) is normally distributed with mean zero and variance t-s.


A d-dimensional Brownian motion is a stochastic process W(t)=(W1(t),,Wd(t)) in d whose coordinate processes Wi(t) are independent one-dimensional Brownian motions.

Figure 1: Sample paths of a standard Brownian motion
Title Brownian motion
Canonical name BrownianMotion
Date of creation 2013-03-22 15:12:46
Last modified on 2013-03-22 15:12:46
Owner skubeedooo (5401)
Last modified by skubeedooo (5401)
Numerical id 16
Author skubeedooo (5401)
Entry type Definition
Classification msc 60J65
Synonym Wiener process
Related topic WienerMeasure
Related topic StochasticCalculusAndSDE