Constructing functions with prescribed pathwise quadratic variation

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.