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

The truncated variation, TV^c${\text{TV}}^c$, is a fairly new concept introduced in ?ochowski (2008) [5]. Roughly speaking, given a càdlàg function f$f$, its truncated variation is “the total variation which does not pay attention to small changes of f$f$, below some threshold c>0$c>0$”. The very basic consequence of such approach is that contrary to the total variation, TV^c${\text{TV}}^c$ 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 ?ochowski (2011) [6], another characterization of TV^c${\text{TV}}^c$ has been found. Namely TV^c${\text{TV}}^c$ is the smallest possible total variation of a function which approximates f$f$ uniformly with accuracy c/2$c/2$. Due to these properties we envisage that TV^c${\text{TV}}^c$ might be a useful concept both in the theory and applications of stochastic processes.