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

Rafał M. Łochowski and Piotr Miłoś

Stochastic Processes and their Applications, 2013, vol. 123, issue 2, 446-474

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

Keywords: Stochastic processes; Semimartingales; Diffusions; Truncated variation; Total variation (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (3)

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DOI: 10.1016/j.spa.2012.08.007

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