Question

In [Föllmer 81] (English translation to be found here) writes: "The class of processes of quadratic variation is clearly larger than the class of semimartingales: Just consider a deterministic process of quadratic variation which is of unbounded variation."

Could anyone please give me examples (with references) of deterministic processes of quadratic variation which are of unbounded variation? Thank you!

(P.S.: What seems to make these deterministic processes interesting is that you also have to use Ito integrals to integrate them)

Answer 1

Take $f:[0,1]\to\mathbb{R}$ such that $f(0)=0$ and it interpolates linearly between $f(1/n)=\frac{(-1)^n}{n}$ for any natural $n$.

Answer 2

By the standard definition in stochastic calculus, with mesh size going to 0, the quadratic variation of the example given is 0, and it is clearly of infinite variation. But does anyone know an example with nonzero finite quadratic variation?