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

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.