Constructing functions with prescribed pathwise quadratic variation

TL;DR

This paper develops methods to construct continuous functions with specific quadratic variation properties, enabling their use in pathwise stochastic calculus and providing deterministic support theorems for diffusions.

Contribution

It introduces new constructions of functions with prescribed quadratic variations, extending the Doss--Sussman method to nonlinear Itô equations in a pathwise setting.

Findings

Constructed functions with prescribed quadratic variations.

Extended Doss--Sussman method for nonlinear pathwise Itô equations.

Provided a deterministic support theorem for diffusions.

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.