On non-negative solutions of SDDEs with an application to CARMA processes

TL;DR

This paper establishes a simple sufficient condition for the non-negativity of solutions to SDDEs driven by subordinators and explores its implications for CARMA processes, extending to multidimensional cases.

Contribution

It introduces a new simple non-negativity condition for SDDEs driven by subordinators and compares its applicability to existing CARMA process results, including multidimensional extensions.

Findings

The condition guarantees non-negativity of SDDE solutions.

It applies to certain CARMA(p,q) processes with p ≥ 3.

The condition extends to multidimensional SDDEs.

Abstract

This note provides a simple sufficient condition ensuring that solutions of stochastic delay differential equations (SDDEs) driven by subordinators are non-negative. While, to the best of our knowledge, no simple non-negativity conditions are available in the context of SDDEs, we compare our result to the literature within the subclass of invertible continuous-time ARMA (CARMA) processes. In particular, we analyze why our condition cannot be necessary for CARMA($p,q$) processes when $p=2$, and we show that there are various situations where our condition applies while existing results do not as soon as $p\geq 3$. Finally, we extend the result to a multidimensional setting.