Probability & Statistics

Properties of Brownian Motion

Brownian motion is fixed by four axioms, a zero start, independent increments, Gaussian increments of variance equal to elapsed time, and continuous paths. These force nowhere-differentiability, self-similarity, the martingale property, and quadratic variation t.

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Define Brownian motion and enumerate some of its properties. It is written BtB_t, or equivalently W(t)W(t) as a Wiener process.

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#The defining properties

A standard Brownian motion (Bt)t≥0(B_t)_{t \ge 0} is pinned down by four conditions.

  • It starts at the origin, B0=0B_0 = 0.
  • It has independent increments, so for s<ts < t the increment Bt−BsB_t - B_s is independent of the entire path up to time ss.
  • Its increments are stationary and Gaussian, Bt−Bs∼N(0,t−s)B_t - B_s \sim \mathcal{N}(0, t-s), depending only on the elapsed time.
  • Its sample paths are continuous.
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The path starts at zero, moves by independent increments, and over any interval the jump is normal with variance equal to the elapsed time. Those few rules, plus continuity, pin Brownian motion down completely.

#What follows

From those few rules flow the striking features that make the process so useful and so strange.

  • The paths are continuous everywhere yet differentiable nowhere, with infinite total variation on every interval.
  • It is self-similar, since BctB_{ct} shares the law of c Bt\sqrt{c}\,B_t.
  • It is at once a martingale and a Markov process, and its quadratic variation over [0,t][0,t] is exactly tt, the single fact that powers Ito calculus.