On nonnegative solutions of SDDEs with an application to CARMA processes
Volume 8, Issue 3 (2021), pp. 309–328
Pub. online: 31 March 2021
Type: Research Article
Open Access
Open Access
Received
16 October 2020
16 October 2020
Revised
12 March 2021
12 March 2021
Accepted
12 March 2021
12 March 2021
Published
31 March 2021
31 March 2021
Abstract
This note provides a simple sufficient condition ensuring that solutions of stochastic delay differential equations (SDDEs) driven by subordinators are nonnegative. While, to the best of our knowledge, no simple nonnegativity 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 $p\ge 3$. Finally, we extend the result to a multidimensional setting.
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