Truncated variation, upward truncated variation and downward truncated variation of Brownian motion with drift — Their characteristics and applications

In Łochowski (2008) [9] we defined truncated variation of Brownian motion with drift, Wt=Bt+μt,t≥0Wt=Bt+μt,t≥0, where (Bt)(Bt) is a standard Brownian motion. Truncated variation differs from regular variation in neglecting jumps smaller than some fixed c>0c>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 interpretations in financial mathematics. The 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 the time parameter of the moment-generating functions of the upward and downward truncated variations.As an application of the formula obtained we give an exact formula for the expected values of upward and downward truncated variations. We also give exact (up to universal constants) estimates of the expected values of the quantities mentioned.