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On Positive Recurrence of One-Dimensional Diffusions with Independent Switching

In Memory of Svetlana Anulova (19.10.1952–21.11.2020)

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 371)

Abstract

Positive recurrence of one-dimensional diffusion with switching, with an additive Wiener process, and with one recurrent and one transient regime is established under suitable conditions on the drift in both regimes and on the intensities of switching. The approach is based on an embedded Markov chain with alternating jumps: one jump increases the average of the square norm of the process, while the next jump decreases it, and under suitable balance conditions this implies positive recurrence.

Keywords

  • 1D diffusion
  • Switching
  • Positive recurrence

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Acknowledgments

The author is deeply indebted to S. Anulova who made useful comments to the first versions of the manuscript and with whom the author started studying the theme of switched diffusions some years ago. This article is devoted to her memory. The article was prepared within the framework of the HSE University Basic Research Program in part which includes Lemmata 1 and 2; the Proof of Lemma 3 and Theorem 1 was funded by Russian Foundation for Basic Research grant 20-01-00575a.

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Correspondence to Alexander Veretennikov .

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Veretennikov, A. (2021). On Positive Recurrence of One-Dimensional Diffusions with Independent Switching. In: Shiryaev, A.N., Samouylov, K.E., Kozyrev, D.V. (eds) Recent Developments in Stochastic Methods and Applications. ICSM-5 2020. Springer Proceedings in Mathematics & Statistics, vol 371. Springer, Cham. https://doi.org/10.1007/978-3-030-83266-7_18

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