If , is a subdivision of the time interval , we denote by , the mesh of this subdivision.
Proposition. Let be a standard Brownian motion. Let . For every sequence of subdivisions such that , the following convergence takes place in (and thus in probability),
As a consequence, almost surely, Brownian paths have an infinite variation on the time interval .
Let us denote
Thanks to the stationarity and the independence of Brownian increments, we have:
Let us now prove that, as a consequence of this convergence, the paths of the process almost surely have an infinite variation on the time interval . It suffices to prove that there exists a sequence of subdivisions such that almost surely
Reasoning by absurd, let us assume that the supremum on all the subdivisions of the time interval of the sums
may be bounded from above by some positive . From the above result, since the convergence in probability implies the existence of an almost surely convergent subsequence, we can find a sequence of subdivisions whose mesh tends to and such that almost surely,
We get then
which is clearly absurd
Exercise. Let be a Brownian motion.
- Show that for , almost surely
- Show that there exists a sequence of subdivisions whose mesh tends to and such that almost surely