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This paper summarizes what have been done in our recent paper [24], which is concerned with stability of a class of switching jump-diffusion processes. The motivation of our study stems from a wide range of applications in communication systems, flexible manufacturing and production planning, financial engineering, and economics. A distinct feature of the system given by (X(t), α(t)) is the switching...
This work focuses on hybrid switching diffusion systems. After presenting the basic models, a difficult and fundamental problem, namely, well posedness is addressed. In particular, continuous and smooth dependence on initial data is treated. The main ideas are presented, whereas the verbatim proofs can be found in [35].
This work is based on our results in. Dealing with two-component Markov processes having continuous dynamics and discrete events, we develop asymptotic properties of regime-switching diffusions. First, strong Feller property is established. Next, classifying the underlying processes as recurrence (positive recurrence or null recurrence) or transience, sufficient conditions for such processes are obtained...
For the pressing needs in control and optimization using hybrid systems, this work focuses on weak stochastic stability and ergodicity of regime-switching diffusions. Using Liapunov functions, we derive necessary and sufficient conditions for weak stability. Then, ergodicity of weakly stable regime-switching diffusions is obtained by constructing cycles using the associated discrete-time Markov chains.
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