Deterministic Integral and Ordinary Differential Equations over Irregular Paths

TL;DR

This paper introduces a deterministic integral for irregular paths, explores its existence, and applies it to solve differential equations driven by such paths, extending classical integration theories to lower H"older exponents.

Contribution

It defines a new integral for irregular paths and studies differential equations driven by these paths, especially for H"older exponents below 1/3, expanding existing integration frameworks.

Findings

Defined a deterministic integral for irregular paths.

Proved existence, uniqueness, and continuity of solutions to irregular path-driven ODEs.

Extended integration theory to H"older exponents less than 1/3.

Abstract

We define a deterministic integral with respect to irregular paths as a limit of standard line integrals and completely describe a class of all paths for which this integral exists for functions with H\"older exponent in the range of (0,1]. With the developed integral calculus, we study the existence, uniqueness and continuity of solution of time- and path-dependent ordinary differential equations driven by irregular paths in traditional H\"older spaces. These results can be viewed as a supplement to the Young-Lyons-Gubinelli integration theory, in particular, for H\"older exponents less than 1/3.