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Jump-diffusions with state-dependent switching: existence and uniqueness, Feller property, linearization, and uniform ergodicity. (English) Zbl 1274.60253
The authors analyze a class of jump-diffusions with state-space dependent switching. In contrast to previous work their model allows for a characteristic measure which is \(\sigma\)-finite. ‘Existence and uniqueness of the underlying process are obtained by representing the switching component as a stochastic integral with respect to a Poisson random measure and by using a successive approximation method’. Furthermore the authors prove the Feller property. To this end they introduce auxiliary processes and they make use of Radon-Nikodym derivatives. Irreducibility and the fact that ‘all compact sets being petite are demonstrated. Based on these results, the uniform ergodicity is established under a general Lyapunov condition. Finally, easily verifiable conditions for uniform ergodicity are established when the jump-diffusions are linearizable with respect to the variable \(x\) (the state variable corresponding to the jump-diffusion component) in a neighborhood of the infinity’. The results are illustrated by some examples.

MSC:
60J75 Jump processes (MSC2010)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60J27 Continuous-time Markov processes on discrete state spaces
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