  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.
