Divergence of the θ-Euler–Maruyama method for neutral stochastic differential equations with unbounded time-dependent delay and Markovian switching when θ∈(0,1]

Aleksandra Petrović and Marija Milošević

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 246, issue C, 449-476

Abstract: In this paper, neutral stochastic differential equations with unbounded time-dependent delay and Markovian switching (NSDEswUTDDMS), where the coefficients grow superlinearly, are considered. The aim is to determine the class of such equations for which the pth absolute moments of the corresponding θ-Euler–Maruyama (θ-EM) method diverge as the step-size tends to zero, when p∈(0,∞) and θ∈(0,1]. This indicates the strong Lp-divergence of the numerical solution at finite time, for p∈[1,+∞), and shows the failure of its numerically weak convergence. The motivation for this paper comes from the paper M. Milošević, Divergence of the backward Euler method for ordinary stochastic differential equations, Numerical Algorithms 82(4) (2019) 1395–1407, where a class of ordinary SDEs with superlinearly growing coefficients is considered. The first part of the paper contains the results of the Lp-divergence based on certain assumptions for the coefficients of the considered equations for each choice of right-continuous Markov chain with finite state space. In the second part of the paper, the new divergence criterion is established, including weaker assumptions on the coefficients of the equations than those in the first part, with a restriction on the choice of the Markov chain. This result leads to a new class of NSDEswUTDDMS for which the mentioned method diverges. Additionally, the main theoretical results are illustrated by an example and numerical simulations.

Keywords: Neutral stochastic differential equations; Unbounded time-dependent delay; Markovian switching; θ–Euler–Maruyama method; Strong Lp-divergence; Superlinear growth conditions (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:246:y:2026:i:c:p:449-476

DOI: 10.1016/j.matcom.2026.01.036

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