Path dependent equations driven by H\"older processes

TL;DR

This paper establishes existence results for path-dependent differential equations driven by H{"o}lder continuous functions, using Young integrals, Schauder's theorem, and allowing unbounded vector fields, with applications to stochastic process trajectories.

Contribution

It provides new existence results for path-dependent equations driven by H{"o}lder processes, accommodating unbounded vector fields and employing Schauder's theorem.

Findings

Existence of solutions for path-dependent equations driven by H{"o}lder functions.

Application of Schauder's theorem in this context.

Handling unbounded vector fields in the analysis.

Abstract

This paper investigates existence results for path-dependent differential equations driven by a H{\"o}lder function where the integrals are understood in the Young sense. The two main results are proved via an application of Schauder theorem and the vector field is allowed to be unbounded. The H{\"o}lder function is typically the trajectory of a stochastic process.