Stability analysis of the θ-method for hybrid neutral stochastic functional differential equations with jumps

Few results seems to be known about the stability of numerical methods for the hybrid neutral stochastic functional differential equations with jumps (also known as the neutral stochastic functional differential equations with Markovian switching and jumps (NSFDEwMJs)). This paper mainly investigates the exponential stability (both the almost sure and the mean-square exponential stability) of the theta-method for NSFDEwMJs. Precisely, it is first illustrated that the trivial solution of the NSFDEwMJ is almost surely and mean-square exponentially stable. It is then shown that the theta-method can preserve the same conclusions of the trivial solution. Numerical examples are demonstrated to illustrate the obtained results. (C) 2021 Elsevier Ltd. All rights reserved.

Automated summary

This paper investigates the exponential stability of the theta-method for neutral stochastic functional differential equations with Markovian switching and jumps. It is shown that the trivial solution is almost surely and mean-square exponentially stable, and the same conclusion holds for the theta-method. Numerical examples are provided to illustrate the obtained results.