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Local Times for Continuous Paths of Arbitrary Regularity

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  • Donghan Kim

    (Columbia University)

Abstract

We study a pathwise local time of even integer order $$p \ge 2$$ p ≥ 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.

Suggested Citation

  • Donghan Kim, 2022. "Local Times for Continuous Paths of Arbitrary Regularity," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2540-2568, December.
    • Handle: RePEc:spr:jotpro:v:35:y:2022:i:4:d:10.1007_s10959-022-01159-z
    DOI: 10.1007/s10959-022-01159-z
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    References listed on IDEAS

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    1. Banner, Adrian D. & Ghomrasni, Raouf, 2008. "Local times of ranked continuous semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 118(7), pages 1244-1253, July.
    2. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    3. Nicolas Perkowski & David J. Promel, 2014. "Local times for typical price paths and pathwise Tanaka formulas," Papers 1405.4421, arXiv.org, revised Apr 2015.
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