A comparison theorem for state-dependent regime-switching diffusion processes is established, which enables us to pathwise-control the evolution of the state-dependent switching component simply by Markov chains. Moreover, a sharp estimate on the stability of Markovian regime-switching processes under the perturbation of transition rate matrices is provided. Our approach is based on elaborate constructions of switching processes in the spirit of Skorokhod’s representation theorem varying according to the problem being dealt with. In particular, this method can cope with switching processes in an infinite state space and not necessarily of birth–death type. As an application, some known results on the ergodicity and stability of state-dependent regime-switching processes can be improved.

中文翻译：

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状态转换过程转移率矩阵扰动下的比较定理和稳定性

建立了状态依赖的机制切换扩散过程的比较定理，这使得我们能够简单地通过马尔可夫链来路径控制状态依赖的切换组件的演化。此外，还提供了对转移率矩阵扰动下马尔可夫政权转换过程稳定性的精确估计。我们的方法是基于根据 Skorokhod 表示定理的精神精心构建的切换过程，根据所处理的问题而变化。特别是，该方法可以应对无限状态空间中的切换过程，而不必是生死类型。作为一种应用，可以改进有关状态依赖的政权切换过程的遍历性和稳定性的一些已知结果。