In the first chapter of "Introduction to Stochastic Calculus with applications" by Klebaner, there is a very brief mention of the existence of functions with zero quadratic variation but infinite (total/linear) variation but no example is given (and I haven't found any). Would the function $\sin\left(\frac{1}{x}\right)$ on the interval $(0,T)$, $T>0$ satisfy these conditions? It is differentiable on its domain so it should have zero quadratic variation and it oscillates infinitely fast near 0 and therefore the linear variation should be unbounded. Is this reasoning correct? What other functions have the mentioned property?
Asked
Active
Viewed 59 times
0
-
I'm almost certain a function of the form $x^p \sin\left(\frac{1}{x^q}\right)$ with $p,q > 0$ (defined to be $0$ at $x=0)$ will work. I'll let you discover an appropriate choice for $p$ and $q$ (probably there are infinitely many such choices for what you want). As for the variation, you can use the method in this MSE answer by making appropriate (direct) series comparison tests with various $p$-series from elementary calculus -- over-estimate using paths of vertical and horizontal segments, under-estimate using only vertical segments. – Dave L. Renfro May 31 '24 at 17:13
-
As for guessing which values of $p, q$ in my previous comment will work, perhaps the 2nd half of my answer to Looking for a function $f$ that is $n$-differentiable, but $f^{(n)}$ is not continuous will help in making a guess. – Dave L. Renfro May 31 '24 at 17:18