Local Times for Continuous Paths of Arbitrary Regularity

Donghan Kim ()
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Donghan Kim: Columbia University

Journal of Theoretical Probability, 2022, vol. 35, issue 4, 2540-2568

Abstract: 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.

Keywords: Pathwise Itô calculus; Pathwise local time; Pathwise Tanaka–Meyer formulas; Ranked functions of arbitrary regularity; Fractional Brownian motion; 60G17; 60G22; 60H05 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-022-01159-z

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