Time-scale separation and state aggregation in singularly perturbed switching diffusions (Q2712236)

The authors consider a 2-component singularly perturbed stochastic process NEWLINE\[NEWLINEY^\varepsilon(t)= (X(t), \gamma^\varepsilon(t)),\quad t\in [0, T],NEWLINE\]NEWLINE where \(X(t)\) is a 1-dimensional diffusion process and \(\gamma^\varepsilon(t)\) is a pure jump process called a switching process. The states of the process \(\gamma^\varepsilon(t)\) are grouped based on the recurrence property. Thus the process \(Y^\varepsilon(t)\) depends on the aggregated switching process. The main result is Proposition 3.3 establishing the weak convergence of \(Y^\varepsilon\) to a switching diffusion Markov process \(\overline Y\) whose ``averaged'' operator is written explicitly.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00050].