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On a class of generalized Takagi functions with linear pathwise quadratic variation

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  • Alexander Schied

Abstract

We consider a class $\mathscr{X}$ of continuous functions on $[0,1]$ that is of interest from two different perspectives. First, it is closely related to sets of functions that have been studied as generalizations of the Takagi function. Second, each function in $\mathscr{X}$ admits a linear pathwise quadratic variation and can thus serve as an integrator in F\"ollmer's pathwise It\=o calculus. We derive several uniform properties of the class $\mathscr{X}$. For instance, we compute the overall pointwise maximum, the uniform maximal oscillation, and the exact uniform modulus of continuity for all functions in $\mathscr{X}$. Furthermore, we give an example of a pair $x,y\in\mathscr{X}$ such that the quadratic variation of the sum $x+y$ does not exist.

Suggested Citation

  • Alexander Schied, 2015. "On a class of generalized Takagi functions with linear pathwise quadratic variation," Papers 1501.00837, arXiv.org, revised Aug 2015.
    • Handle: RePEc:arx:papers:1501.00837
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    References listed on IDEAS

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    1. Schied, Alexander, 2014. "Model-free CPPI," Journal of Economic Dynamics and Control, Elsevier, vol. 40(C), pages 84-94.
    2. Bick, Avi & Willinger, Walter, 1994. "Dynamic spanning without probabilities," Stochastic Processes and their Applications, Elsevier, vol. 50(2), pages 349-374, April.
    3. Mark Davis & Jan Obłój & Vimal Raval, 2014. "Arbitrage Bounds For Prices Of Weighted Variance Swaps," Mathematical Finance, Wiley Blackwell, vol. 24(4), pages 821-854, October.
    4. Alexander Schied, 2013. "Model-free CPPI," Papers 1305.5915, arXiv.org, revised Jan 2014.
    5. Hans Follmer & Alexander Schied, 2013. "Probabilistic aspects of finance," Papers 1309.7759, arXiv.org.
    6. Christian Bender & Tommi Sottinen & Esko Valkeila, 2008. "Pricing by hedging and no-arbitrage beyond semimartingales," Finance and Stochastics, Springer, vol. 12(4), pages 441-468, October.
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