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On the entrance at infinity of Feller processes with no negative jumps

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  • Foucart, Clément
  • Li, Pei-Sen
  • Zhou, Xiaowen

Abstract

Consider a non-explosive positive Feller process with no negative jumps. It is shown in this note that when infinity is an entrance boundary, in the sense that the entrance times of the process remain bounded when the initial value tends to infinity, the process admits a Feller extension on the compactified state space [0,∞]. Moreover, when started from infinity, the extended Markov process on [0,∞] leaves infinity instantaneously and stays finite, almost-surely. Arguments are adapted from a proof given by Kallenberg (2002) for diffusions. We also show that the process started from x converges weakly towards that started from infinity in the Skorokhod space, when x goes to infinity.

Suggested Citation

  • Foucart, Clément & Li, Pei-Sen & Zhou, Xiaowen, 2020. "On the entrance at infinity of Feller processes with no negative jumps," Statistics & Probability Letters, Elsevier, vol. 165(C).
    • Handle: RePEc:eee:stapro:v:165:y:2020:i:c:s0167715220301620
    DOI: 10.1016/j.spl.2020.108859
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    References listed on IDEAS

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    1. Fu, Zongfei & Li, Zenghu, 2010. "Stochastic equations of non-negative processes with jumps," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 306-330, March.
    2. Li, Pei-Sen, 2019. "A continuous-state polynomial branching process," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2941-2967.
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    Cited by:

    1. Yong-Hua Mao & Tao Wang, 2022. "Convergence Rates in Uniform Ergodicity by Hitting Times and $$L^2$$ L 2 -Exponential Convergence Rates," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2690-2711, December.

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