Delay equations with non-negativity constraints driven by a H\"older continuous function of order \beta in (1/3,1/2)

TL;DR

This paper establishes existence, uniqueness, and bounds for solutions of multidimensional delay differential equations with non-negativity constraints driven by H"older continuous functions of order between 1/3 and 1/2, including applications to fractional Brownian motion.

Contribution

It provides the first known results on existence and uniqueness for such delay equations with non-negativity constraints driven by H"older noise in the specified range.

Findings

Proved existence and uniqueness of solutions.

Derived bounds for the supremum norm of solutions.

Applied results to fractional Brownian motion with Hurst parameter in (1/3, 1/2).

Abstract

In this note we prove an existence and uniqueness result of solution for multidimensional delay differential equations with normal reflection and driven by a H\"older continuous function of order $\beta \in (\frac13,\frac12)$. We also obtain a bound for the supremum norm of this solution. As an application, we get these results for stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H $\in (\frac13,\frac12)$.