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Local Times for Continuous Paths of Arbitrary Regularity

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Abstract

We study a pathwise local time of even integer order \(p \ge 2\), defined as an occupation density, for continuous functions with finite pth variation along a sequence of time partitions. With this notion of local time and a new definition of the Föllmer integral, we establish Tanaka-type change-of-variable formulas in a pathwise manner. We also derive some identities involving this high-order pathwise local time, each of which generalizes a corresponding identity from the theory of semimartingale local time. We then use collision local times between multiple functions of arbitrary regularity to study the dynamics of ranked continuous functions.

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Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Acknowledgements

The author would like to thank Professor Ioannis Karatzas for suggesting this topic and for numerous discussions regarding the material in this paper, and Professors Rama Cont and Nicolas Perkowski for helpful comments and for correcting errors in an earlier version of this paper. Thanks also go to Professors Hans Föllmer and Nicolas Perkowski for sending us the unpublished thesis of Würmli.

Funding

This study was supported by the Department of Mathematics at Columbia University.

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  1. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    Donghan Kim

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  1. Donghan Kim
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Correspondence to Donghan Kim.

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Kim, D. Local Times for Continuous Paths of Arbitrary Regularity. J Theor Probab 35, 2540–2568 (2022). https://doi.org/10.1007/s10959-022-01159-z

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