Our aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

中文翻译：

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随机积分关于马尔可夫链的状态占有测度的弱收敛

我们的目标是找到随机积分相对于马尔可夫链的状态占有量度的弱收敛的充分条件。首先，我们研究了马尔可夫链的状态指示函数和状态占用度量的性质。特别是，我们在生成矩阵的缩放下建立了状态占用度量的弱收敛。然后，依靠状态占用测度与 Dynkin 鞅之间的联系，我们为状态占用测度的随机积分的弱收敛提供了充分条件。我们应用我们的结果来推导出马尔可夫调制 Erlang 损失模型和状态切换 Cox-Ingersoll-Ross 过程的扩散极限。