On the Nonexplosion and Explosion for Nonhomogeneous Markov Pure Jump Processes
In this paper, we obtain new drift-type conditions for nonexplosion and explosion for nonhomogeneous Markov pure jump processes in Borel state spaces. The conditions are sharp; e.g., the one for nonexplosion is necessary if the state space is in addition locally compact and the Q-function satisfies weak Feller-type and local boundedness conditions. We comment on the relations of our conditions with the existing ones in the literature and demonstrate some possible applications.
KeywordsDynkin’s formula Nonhomogeneous Markov pure jump process Nonexplosion
Mathematics Subject Classification (2010)60J75 90C40
I would like to thank Professor Mufa Chen (Beijing Normal University) for providing the scan copy of the relevant pages in  and the paper . I also thank the referees for their helpful comments and remarks. This work was carried out with a financial grant from the Research Fund for Coal and Steel of the European Commission, within the INDUSE-2-SAFETY project (Grant No. RFSR-CT-2014-00025).
- 6.Chen, M.: A comment on the book “Continuous-time Markov Chains” by W. J. Anderson. Chin. J. Appl. Probab. Stat. 12, 55–59 (1996). Updated version also available at arXiv:1412.5856
- 9.Chen, M.: Practical criterion for uniqueness of q-processes. Chin. J. Appl. Probab. Stat. 31, 213–224 (2015). arXiv:1503.02119
- 10.Chow, P., Khasminskii, R.: Methods of Lyapunov functions for analysis of absorption and explosion in Markov chains. Probl. Inf. Transm. 47, 232–250 (2011). (Translation of the original Russian text at Problemy Peredachi Informatsii 47, 19–38 (2011))Google Scholar
- 13.Feinberg, E., Mandava, M., Shiryaev, A.: Kolmogorov’s equations for jump Markov processes with unbounded jump rates (2016). arXiv:1603.02367
- 16.Hu, D.: Analytic Theory of Markov Processes in General State Spaces. Wuhan University Press, Wuhan (2013). Republication of the edition originally published in 1985 (in Chinese) Google Scholar
- 22.Piunovskiy, A., Zhang, Y.: Discounted continuous-time markov decision processes with unbounded rates and randomized history-dependent policies: the dynamic programming approach. 4OR-Q. J. Oper. Res. 12, 49–75 (2014). The extended version is available at arXiv:1103.0134
- 30.Zheng, J.: Phase Transitions of Ising Model on Lattice Fractals, Martingale Approach for \(q\)-Processes. Ph.D. thesis. Beijing Normal University (1993) (in Chinese) Google Scholar