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

Rafał M. Łochowski

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On Truncated Variation of Brownian Motion with Drift

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We introduce the concept of truncated variation of Brownian motion with drift, which differs from regular variation by neglecting small jumps (smaller than some c > 0). We estimate the expected value of the truncated variation. The behaviour resembling phase transition as c varies is revealed. Truncated variation appears in the formula for an upper bound for return from any trading based on a single asset with flat commission.

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