Stability of a class of neutral stochastic differential equations with unbounded delay and Markovian switching and the Euler–Maruyama method

This paper is motivated by the paper Hu et al. (2013). This paper contains the existence and uniqueness, as well as stability results of the exact solution for a class of neutral stochastic differential equations with unbounded delay and Markovian switching, including the case when the delay function is bounded. Moreover, the convergence in probability of the Euler–Maruyama method is established regardless whether or not the delay function is bounded. These results are obtained under certain non-linear growth conditions on the coefficients of the equation. Adding the linear growth condition on the drift coefficient, the almost sure exponential stability of the Euler–Maruyama method is proved in the case of the bounded delay. The presence of the neutral term is essential for consideration of this class of equations. It should be stressed that the neutral term is also hybrid, that is, it depends on the Markov chain. Moreover, the Euler–Maruyama method is defined in a non-trivial way regarding the neutral term.