Truncated variation, upward truncated variation and downward truncated variation of Brownian motion with drift—their mean value and their applications
Volume 90 / 2010
Abstract
In \cite{L 2008} for $c>0$ we defined the truncated variation, $TV_{\mu}^{c},$ of a Brownian motion with drift, $W_t = B_t + \mu t, t\geq 0,$ where $(B_t)$ is a standard Brownian motion. In this article we define two related quantities: the upward truncated variation $$ UTV^c_{\mu}[a,b]=\sup_n \sup_{a \leq t_1 < s_1 < \ldots < t_n < s_n \leq b} \sum_{i=1}^{n} \max \{ W_{s_i} - W_{t_i} - c, 0 \} $$ and, analogously, the downward truncated variation $$ DTV^c_{\mu}[a,b]=\sup_n \sup_{a \leq t_1 < s_1 < \ldots < t_n < s_n \leq b} \sum_{i=1}^{n} \max \{ W_{t_i} - W_{s_i} - c, 0 \}. $$ 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_{\mu}^c $ 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.
