One-dimensional game-theoretic differential equations

TL;DR

This paper introduces a framework for solving one-dimensional differential equations driven by typical paths using Itô type integration, establishing existence, uniqueness, and approximation results even with non-Lipschitz coefficients.

Contribution

It develops a novel approach for differential equations driven by typical paths, extending classical results to non-Lipschitz coefficients within a robust Itô integration framework.

Findings

Proves existence and uniqueness of solutions for the equations.

Establishes approximation results similar to Doss--Sussmann.

Extends classical theory to non-Lipschitz coefficients.

Abstract

We provide a very brief introduction to typical paths and the corresponding It\^o type integration. Relying on this robust It\^o integration, we prove an existence and uniqueness result for one-dimensional differential equations driven by typical paths with non-Lipschitz continuous coefficients in the spirit of Yamada--Watanabe as well as an approximation result in the spirit of Doss--Sussmann.