@MISC{Miłoś_exactrepresentation, author = {Piotr Miłoś}, title = {EXACT REPRESENTATION OF TRUNCATED VARIATION OF BROWNIAN MOTION}, year = {} }

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Abstract

Abstract. In the recent papers [8, 9, 10] 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 Lévy subordinator with the exponent 2q tanh(c q/2). This also gives a representation of a solution of the two-sided Skorokhod problem for a Brownian motion. 1. Representation of the truncated variation Given f: [a; b] 7 → R, a càdlàg function, its truncated variation on the interval [a; b] is defined by (1.1) TVc(f, [a; b]): = sup n sup a≤t1<t2<...<tn≤b n−1∑ i=1 φc (f(ti+1) − f(ti)) , c ≥ 0, φc (x) = max {|x | − c, 0} (note that TV0(f, [a; b]) is the total variation). Closely related notions of the upward truncated variation and the downward truncated variation denoted by UTVc, respectively DTVc are defined by putting φc(x) = max {x − c, 0}, respectively φc(x) = max {−x − c, 0} in (1.1). The truncated variation has two interesting variational characterizations- [9, (1.2) and (2.2)]. We recall one of them: (1.2) TVc(f, [a; b]) = inf TV0(g, [a; b]) : g such that ‖g − f‖ ∞ ≤