Truncated variation of a stochastic process - its optimality for processes with cadlag trajectories and its limit distributions for diffusions

Rafal Lochowski (Warschau)
Wednesday, April 18, 2012 - 5:00pm
TU Berlin, MA041, Straße des 17. Juni 136, 10623 Berlin

For a given function f : [a; b] − > RI define its truncated variation at the level c >0 with the following formula

TV^c (f , [a; b]) := sup_{n} sup_ {a ≤t_1 < t_2 <...< t_n ≤b} \sum^{n−1}_{i=1} max {|f (t_{i+1} ) − f (t_{i} )| − c, 0}. For f being a cadlag function its truncated variation is always finite, in opposite to total variation, which is a limit value of TV^c (f , [a; b]) as c tends to 0 and may be infinite. I will prove that TV^c (f , [a; b]) is the smallest possible and attainable total variation of a function uniformly approximating f with accuracy c/2. This may be viewed as a generalization of Hahn-Jordan decomposition of a function with finite total variation. In the second part of my talk I will present results on limit distributions of truncated variation process of diffusions as c tends to 0.