Author
Cont, R
Perkowski, N
Journal title
Transactions of the American Mathematical Society
DOI
10.1090/btran/34
Volume
6
Last updated
2024-03-13T04:46:50.01+00:00
Page
161-161
Abstract
We construct a pathwise integration theory, associated with a change of
variable formula, for smooth functionals of continuous paths with arbitrary
regularity defined in terms of the notion of $p$-th variation along a sequence
of time partitions. For paths with finite $p$-th variation along a sequence of
time partitions, we derive a change of variable formula for $p$ times
continuously differentiable functions and show pointwise convergence of
appropriately defined compensated Riemann sums. Results for functions are
extended to regular path-dependent functionals using the concept of vertical
derivative of a functional. We show that the pathwise integral satisfies an
`isometry' formula in terms of $p$-th order variation and obtain a `signal plus
noise' decomposition for regular functionals of paths with strictly increasing
$p$-th variation. For less regular ($C^{p-1}$) functions we obtain a
Tanaka-type change of variable formula using an appropriately defined notion of
local time. These results extend to multidimensional paths and yield a natural
higher-order extension of the concept of `reduced rough path'. We show that,
while our integral coincides with a rough-path integral for a certain rough
path, its construction is canonical and does not involve the specification of
any rough-path superstructure.
Symplectic ID
867305
Download URL
http://arxiv.org/abs/1803.09269v2
Favourite
On
Publication type
Journal Article
Publication date
10 Apr 2019
Please contact us with feedback and comments about this page. Created on 10 Jul 2018 - 17:40.