On truncated variation, upward truncated variation and downward truncated variation for diffusions

The truncated variation, TVcTVc, is a fairly new concept introduced in Łochowski (2008) [5]. Roughly speaking, given a càdlàg function ff, its truncated variation is “the total variation which does not pay attention to small changes of ff, below some threshold c>0c>0”. The very basic consequence of such approach is that contrary to the total variation, TVcTVc is always finite. This is appealing to the stochastic analysis where so-far large classes of processes, like semimartingales or diffusions, could not be studied with the total variation. Recently in Łochowski (2011) [6], another characterization of TVcTVc has been found. Namely TVcTVc is the smallest possible total variation of a function which approximates ff uniformly with accuracy c/2c/2. Due to these properties we envisage that TVcTVc might be a useful concept both in the theory and applications of stochastic processes.For this reason we decided to determine some properties of TVcTVc for some well-known processes. In course of our research we discover intimate connections with already known concepts of the stochastic processes theory.First, for semimartingales we proved that TVcTVc is of order c−1c−1 and the normalized truncated variation converges almost surely to the quadratic variation of the semimartingale as c↘0c↘0. Second, we studied the rate of this convergence. As this task was much more demanding we narrowed to the class of diffusions (with some mild additional assumptions). We obtained the weak convergence to a so-called Ocone martingale. These results can be viewed as some kind of law of large numbers and the corresponding central limit theorem.Finally, for a Brownian motion with a drift we proved the behavior of TVcTVc on intervals going to infinity. Again, we obtained a LLN and CLT, though in this case they have a different interpretation and were easier to prove.All the results above were obtained in a functional setting, viz. we worked with processes describing the growth of the truncated variation in time. Moreover, in the same respect we also treated two closely related quantities—the so-called upward truncated variation and downward truncated variation.