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## Banach Center Publications

2010 | 90 | 1 | 67-78
Tytuł artykułu

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

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In [L 2008] for c > 0 we defined the truncated variation, $TV_{μ}^{c}$, of a Brownian motion with drift, $W_t = B_t + μt$, t ≥ 0, where $(B_t)$ is a standard Brownian motion. In this article we define two related quantities: the upward truncated variation <br> $UTV^{c}_{μ}[a,b] = sup_{n} sup_{a≤t₁<s₁<...<tₙ<sₙ≤b} ∑_{i=1}^{n} max{W_{s_i} - W_{t_i} - c, 0}$ <br> and, analogously, the downward truncated variation <br> $DTV^{c}_{μ}[a,b] = sup_{n} sup_{a≤t₁<s₁<...<tₙ<sₙ≤b} ∑_{i=1}^{n} max{W_{t_i} - W_{s_i} - c, 0}$. <br> 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 $UTV_{μ}^{c}$ 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.
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Tom
Numer
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67-78
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wydano
2010
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• Department of Mathematics and Mathematical Economics, Warsaw School of Economics, Al. Niepodległości 162, 02-554 Warszawa, Poland
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