In this paper a class of neutral stochastic differential equations with unbounded delay and Markovian switching, where the coefficients grow superlinearly, is studied. It is proved that the corresponding Euler–Maruyama approximation converges neither in the strong \( L^p \) -sense nor in the numerically weak sense to the exact solution at a finite time point, for all \( p \in (0,\infty ). \) This research can be regarded as an extension of the results presented in the paper Hutzenthaler et al. (Proc. R. Soc. A 467(2130), 1563–1576 467, 2011), where the class of ordinary stochastic differential equations is considered. In the first part of the paper, the \( L^p \) -divergence criterion is based on certain assumptions for the coefficients of the considered equations for each choose of right-continuous Markov chain with finite space set. In the second part of the paper, the new divergence criteria is established, which includes weaker assumptions on the coefficients of the equations than in the first part of the paper, while there is a restriction in the choice on the Markov chain. In that way, a new class of neutral stochastic differential equations with unbounded delay and Markovian switching for which the mentioned method diverges is determined. Additionally, to support the main result of the paper an example and numerical simulations are provided. It is shown that a class of considered equations which have unique global solutions with finite first order absolute moments and infinite first order absolute moments of the Euler–Maruyama solutions, at finite time, is not empty.
