# Truncated Variation, Upward Truncated Variation and Downward Truncated Variation of Brownian Motion with Drift - their Characteristics and Applications - Mathematics > Probability

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Abstract: In the paper -On Truncated Variation of Brownian Motion with Drift- Bull.Pol. Acad. Sci. Math. 56 2008, no.4, 267 - 281 we defined truncatedvariation of Brownian motion with drift, $W t = B t + \mu t, t\geq 0,$ where$B t$ is a standard Brownian motion. Truncated variation differs from regularvariation by neglecting jumps smaller than some fixed $c > 0$. We prove thattruncated variation is a random variable with finite moment-generating functionfor any complex argument. We also define two closely related quantities -upward truncated variation and downward truncated variation. The definedquantities may have some interpretation in financial mathematics. Exponentialmoment of upward truncated variation may be interpreted as the maximal possiblereturn from trading a financial asset in the presence of flat commission whenthe dynamics of the prices of the asset follows a geometric Brownian motionprocess. We calculate the Laplace transform with respect to time parameter ofthe moment-generating functions of the upward and downward truncatedvariations. As an application of the obtained formula we give an exact formulafor expected value of upward and downward truncated variations. We give alsoexact up to universal constants estimates of the expected values of thementioned quantities.

Autor: Rafał Łochowski

Fuente: https://arxiv.org/