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. For positive c we define two related quantities - upward truncated variation UTV^c_{\mu[a,b]=\sup_n a \leq t_1 < s_1 < ... < t_n < s_n \leq b i=1^{n} \max }%DIFDELCMD < {%%% W_{s_i and , analogously, downward truncated variation DTV^c_{\mu[a,b]=\sup_n a \leq t_1 < s_1 < ... < t_n < s_n \leq b i=1^{n} \max }%DIFDELCMD < {%%% W_{t_i We prove that exponential moments of the above quantities are finite (in opposite to the regular variation, corresponding to TV^0, which is infinite almost surely). We present estimates of the expected value of \% UTV^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 Browniam motion process. In the presented estimates upward truncated variation appears naturally .
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 [a,b]=\sup_n a \leq t_1 < s_1 < ... < t_n < s_n \leq b i=1^{n} \max }%DIFDELCMD < {%%% and downward truncated variation [a,b]=\sup_n a \leq t_1 < s_1 < ... < t_n < s_n \leq b i=1^{n} \max }%DIFDELCMD < {%%% . 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 .
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For positive c we define two
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
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UTV^c_{\mu
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and , analogously,
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DTV^c_{\mu
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We prove that exponential moments of the above quantities are finite (in opposite to the regular variation, corresponding to TV^0, which is infinite almost surely). We present estimates of the expected value of \% UTV^c up to universal constants. As an application we give some estimates of the maximal possible gain
. The defined quantities may have some interpretation in financial mathematics. Exponential moment of upward truncated variation may be interpreted as the maximal possible return
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(proportional to the value of the transaction)
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Browniam
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In the presented estimates upward truncated variation appears naturally
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