Brownian Motion, 2nd Edition

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 [t_j-1, t_j] the oscillation of a continuous function ƒ should roughly be |ƒ(t_j) - ƒ (t_j-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: ...