Constructing functions with prescribed pathwise quadratic variation
Probability
2019-07-02 v3 Classical Analysis and ODEs
Abstract
We construct rich vector spaces of continuous functions with prescribed curved or linear pathwise quadratic variations. We also construct a class of functions whose quadratic variation may depend in a local and nonlinear way on the function value. These functions can then be used as integrators in F\"ollmer's pathwise It\=o calculus. Our construction of the latter class of functions relies on an extension of the Doss--Sussman method to a class of nonlinear It\=o differential equations for the F\"ollmer integral. As an application, we provide a deterministic variant of the support theorem for diffusions. We also establish that many of the constructed functions are nowhere differentiable.
Cite
@article{arxiv.1511.04678,
title = {Constructing functions with prescribed pathwise quadratic variation},
author = {Yuliya Mishura and Alexander Schied},
journal= {arXiv preprint arXiv:1511.04678},
year = {2019}
}