Intuitively, if increments become infinitesimally small, why doesn’t Brownian motion become a differentiable function?

  • 2
    $\begingroup$ Similar question: why should it? $\endgroup$ – Randall Mar 12 at 23:01
  • $\begingroup$ Have you heard of fractals? $\endgroup$ – Don Thousand Mar 12 at 23:04
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    $\begingroup$ Increments becoming small is continuity. Continuity does not imply differentiability. $\endgroup$ – Kavi Rama Murthy Mar 12 at 23:07
  • $\begingroup$ In a small time interval $t$, the BM will typically move $\simeq t^{1/2}$ (the standard deviation), which is too much to make it differentiable. $\endgroup$ – user138530 Mar 12 at 23:30
  • $\begingroup$ Even for a stable levy process (for instance) the increments become “infinitesimally small” in the sense that the small increments converge in probability to zero, but in this case the sample paths even fail to be continuous, let alone differentiate. $\endgroup$ – Shalop Mar 12 at 23:32

Imagine a particle moving around on some trajectory.

Its trajectory being continuous means that as you slow time down, the particle stays closer and closer to where it was: no big jumps.

Its trajectory being differentiable means that as you slow time down, the particle doesn't just stay near where it was, it moves more and more in a straight line.

Differentiability is a much, much stronger condition than mere continuity. As you take a limit in Brownian motion, you get a continuous function -- but you have no guarantees on its direction, which is what you need for differentiability.

  • $\begingroup$ This is a great answer to get a visual idea of what is going on here. Thanks! $\endgroup$ – xiA 2 days ago

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