Exact representation of truncated variation of Brownian motion

Abstract

In the recent papers [Lochowski:2011fk, Lochowski:2013yq, Lochowski:2013lr] the truncated variation has been introduced, characterized and studied in various stochastic settings. In this note we uncover an intimate link to the Skorokhod problem. Further, we exploit it to give an explicit representation of the truncated variation of a Brownian motion. More precisely, we prove that the inverse of this process is, up to a minor time shift, a L\'evy subordinator with the exponent \sqrt{2q}\tanh(c\sqrt{q/2}) . This also gives a representation of a solution of the two-sided Skorokhod problem for a Brownian motion.