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In [6] for c > 0 we defined the truncated variation, TV c μ , of a Brownian motion with drift, Wt = Bt + μt, t ≥ 0, where (Bt) is a standard Brownian motion. In this article we define two related quantities: the upward truncated variation UTV c μ [a, b] = sup n sup a≤t1<s1<...<tn<sn≤b n X i=1 max{Wsi −Wti − c, 0} and, analogously, the downward truncated variation DTV c μ [a, b] = sup n sup a≤t1<s1<...<tn<sn≤b n X i=1 max{Wti −Wsi − c, 0}. We prove that the exponential moments of the above quantities are finite (in contrast to the regular variation, corresponding to c = 0, which is infinite almost surely). We present estimates of the expected value of UTV c μ up to universal constants. As an application we give some estimates of the maximal possible gain from trading a financial asset in the presence of flat commission (proportional to the value of the transaction) when the dynamics of the prices of the asset follows a geometric Brownian motion process. In the presented estimates the upward truncated variation appears naturally.

banach center publications

Truncated variation, upward truncated variation and downward truncated variation of Brownian motion with drift–-their mean value and their applications