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: ...
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