Exponential stability of the exact solutions and $\theta$-EM   approximations to neutral SDDEs with Markov switching
  Exponential stability of the exact solutions as well as $\theta$-EM ($\frac{1}{2}<\theta\le 1$) approximations to neutral stochastic differential delay equations with Markov switching will be investigated in this paper. Sufficient conditions are obtained to ensure the $p$-th moment ($p\ge1$) and almost sure exponential stability of the exact solutions as well as $\theta$-EM approximations ($p=2$). An example will be presented to support our conclusions. 
Guangqiang Lan and Chenggui Yuan
math.PR