No simple arbitrage for fractional Brownian motion

We prove the following result: For $(Z_t)_{t \in \mathbf{R}}$ a fractional Brownian motion with arbitrary Hurst parameter, there does not exist any stopping time $τ$ adapted to the natural filtration of the increments of $Z$ such that, with positive probability, $τ$ a local minimum at right of the trajectory of $Z$.

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