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

TL;DR

This paper establishes a precise equivalence between blow-up solutions of stochastic delay differential equations and their non-delayed versions, with implications for biology and finance models.

Contribution

It proves that SDDEs with a single delay blow up if and only if their instantaneous counterparts do, extending previous results with a comparison theorem approach.

Findings

Blow-up solutions in SDDEs are equivalent to those in their non-delayed forms.

The result applies to models in biology and finance.

The proof uses a comparison theorem by Ikeda and Watanabe.

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.