Stochastic Delay Differential Equations have blow-up solutions if and only if their instantaneous counterparts have them

Abstract

Motivated by a recent publication by Ishiwata and Nakata (2022), we prove that sufficiently regular stochastic delay differential equations (SDDEs) with a single discrete delay have blow up solutions if and only if their undelayed counterparts have them, using a comparison theorem by Ikeda and Watanabe (1977). This result has applications in mathematical biology and finance.