On limit distributions of normalized truncated variation, upward truncated variation and downward truncated variation processes

Abstract: In the paper we introduce the truncated variation, upward truncated variation and downward truncated variation. These are closely related to the total variation but are well-defined even if the latter is infinite. Our aim is to explore their feasibility to studies of stochastic processes. We concentrate on a Brownian motion with drift for which we prove the convergence of the above- mentioned quantities. For example, we study the truncated variation when the truncation parameter c tends to 0. We prove in this case that for "small" c's it is well-approximated by a deterministic process. Moreover we prove that error in this approximation converges weakly (in functional sense) to a Brownian motion. We prove also similar result for truncated variation processes when time parameter is rescaled to infinity. We stress that our methodology is robust. A key to the proofs was a decomposition of the truncated variation (see Lemmas 11 and 12). It can be used for studies of any continuous processes. Some additional results like an analog of the Anscombe-Donsker theorem and the Laplace transform of time to given drawdown by c (and analogously drawup till time) are presented.