Pathwise stochastic calculus with local times
File(s)1508.05984v1.pdf (431.19 KB)
Accepted version
Author(s)
Davis, M
Obłój, J
Siorpaes, P
Type
Journal Article
Abstract
We study a notion of local time for a continuous path, defined as a limit of
suitable discrete quantities along a general sequence of partitions of the time
interval. Our approach subsumes other existing definitions and agrees with the
usual (stochastic) local times a.s. for paths of a continuous semimartingale.
We establish pathwise version of the It\^o-Tanaka, change of variables and
change of time formulae. We provide equivalent conditions for existence of
pathwise local time. Finally, we study in detail how the limiting objects, the
quadratic variation and the local time, depend on the choice of partitions. In
particular, we show that an arbitrary given non-decreasing process can be
achieved a.s. by the pathwise quadratic variation of a standard Brownian motion
for a suitable sequence of (random) partitions; however, such degenerate
behavior is excluded when the partitions are constructed from stopping times.
suitable discrete quantities along a general sequence of partitions of the time
interval. Our approach subsumes other existing definitions and agrees with the
usual (stochastic) local times a.s. for paths of a continuous semimartingale.
We establish pathwise version of the It\^o-Tanaka, change of variables and
change of time formulae. We provide equivalent conditions for existence of
pathwise local time. Finally, we study in detail how the limiting objects, the
quadratic variation and the local time, depend on the choice of partitions. In
particular, we show that an arbitrary given non-decreasing process can be
achieved a.s. by the pathwise quadratic variation of a standard Brownian motion
for a suitable sequence of (random) partitions; however, such degenerate
behavior is excluded when the partitions are constructed from stopping times.
Date Issued
2018-02-01
Date Acceptance
2016-09-05
Citation
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2018, 54 (1), pp.1-21
ISSN
0246-0203
Publisher
Elsevier Masson
Start Page
1
End Page
21
Journal / Book Title
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
Volume
54
Issue
1
Copyright Statement
© 2018 Institute Henri Poincaré
Identifier
http://arxiv.org/abs/1508.05984v1
Subjects
Science & Technology
Physical Sciences
Statistics & Probability
Mathematics
Pathwise local-time
Ito-Tanaka formula
Random partitions
Brownian variation
BROWNIAN-MOTION
TANAKA FORMULA
INTEGRATION
PATHS
math.PR
math.PR
60G17, 60H05
Statistics & Probability
0104 Statistics
Notes
30 pages
Publication Status
Published
Date Publish Online
2018-02-19