  In the paper "On Truncated Variation of Brownian Motion with Drift" (Bull.
Pol. Acad. Sci. Math. 56 (2008), no.4, 267 - 281) we defined truncated
variation 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 regular
variation by neglecting jumps smaller than some fixed $c > 0$. We prove that
truncated variation is a random variable with finite moment-generating function
for any complex argument. We also define two closely related quantities -
upward truncated variation and downward truncated variation. The defined
quantities may have some interpretation in financial mathematics. Exponential
moment of upward truncated variation may be interpreted as the maximal possible
return from trading a financial asset in the presence of flat commission when
the dynamics of the prices of the asset follows a geometric Brownian motion
process. We calculate the Laplace transform with respect to time parameter of
the moment-generating functions of the upward and downward truncated
variations. As an application of the obtained formula we give an exact formula
for expected value of upward and downward truncated variations. We give also
exact (up to universal constants) estimates of the expected values of the
mentioned quantities.
