Implicit numerical methods for neutral stochastic differential equations with unbounded delay and Markovian switching

摘要

This paper contains results on the backward Euler method for a class of neutral stochastic differential equations with both unbounded and bounded delays and Markovian switching. The convergence in probability of the backward Euler method is proved under nonlinear growth conditions including the one-sided Lipschitz condition in order for the backward Euler method to be well defined. The presence of the neutral term, which is hybrid, that is, it depends on the Markov chain, is essential for consideration of these equations. It is proved that the discrete backward Euler equilibrium solution is globally a.s. asymptotically exponentially stable without the linear growth condition on the drift coefficient.