9 The variation of Brownian paths

In this chapter we start our study of the regularity of the Brownian paths. One possibility to measure the regularity of a function is to look at its oscillations.

On a small interval [tj-1, tj] the oscillation of a continuous function ƒ should roughly be |ƒ(tj) - ƒ (tj-1) |and summing up the oscillation over consecutive intervals should give a number quantifying the oscillatory behaviour of f. In this way we would measure the increase of a monotone function and, for a Lipschitz continuous functions, the oscillations would be bounded by the Lipschitz constant times the interval length. In order to capture Hölder continuity we will introduce a weight on the oscillations.

Recall the definition of p-variation: ...

Get Brownian Motion, 2nd Edition now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.