[Math] Why isn’t Brownian motion differentiable

brownian motionderivativesdifferential-formsstochastic-calculusstochastic-processes

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

Best Answer

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.

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