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Moments and distributions of the last exit times for a class of Markov processes. (English) Zbl 07316548

Summary: In this paper, we consider some related questions to last exit time of general Markov processes. An estimate for the distribution of the last exit time is derived. An equivalent characterization for the finiteness of the \(k\)-moments of the last exit time is obtained. Finally, some examples are provided to show the significance and usefulness of our results.

MSC:

37Nxx Applications of dynamical systems
93Bxx Controllability, observability, and system structure
92Bxx Mathematical biology in general
60Jxx Markov processes
93Exx Stochastic systems and control
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[1] Bao, H.; Cao, J., Stochastic global exponential stability for neutral-type impulsive neural networks with mixed time-delays and Markovian jumping parameters, Commun. Nonlinear Sci. Numer. Simul., 16, 3786-3791 (2011) · Zbl 1227.34079
[2] Blumenthal, R.; Getoor, R., Markov Processes and Potental Theory (1968), Academy Press: Academy Press New York · Zbl 0169.49204
[3] Chen, D., The potential of Bessel processes and related problems, Acta Math. Sinica, 28, 536-544 (1985) · Zbl 0585.60076
[4] Chung, K., Probabilistic approach in potential theory to the equilibrium problem, Ann. Inst. Fourier Grenoble, 23, 3, 313-322 (1973) · Zbl 0258.31012
[5] Getoor, R., Last exit times and additive functions, Ann. Probab., 1, 550-569 (1973) · Zbl 0324.60062
[6] Getoor, R., The Brownian escape process, Ann. Probab., 7, 864-867 (1979) · Zbl 0416.60086
[7] Hawkes, J., Moments of last exit times, Mathematica, 24, 266-269 (1977) · Zbl 0384.60053
[8] Li, B.; Liu, L., An estimate on the distribution and moments of the last exit time of an elliptic diffusion process, Acta Math. Sci., 26, 4, 639-645 (2006) · Zbl 1137.60336
[9] Li, C.; Wu, R.; Liao, M., The equilibrium measure and the last exit distribution, Chinese J. Appl. Probab. Statist., 9, 289-295 (1993) · Zbl 1001.60511
[10] Pardo, J., On the future infimum of positive self-similar Markov processes, Stoch. Stoch. Rep., 78, 3, 123-155 (2006) · Zbl 1100.60018
[11] Pardo, J., The upper envelope of positive self-similar Markov processes, J. Theoret. Probab., 22, 514-542 (2009) · Zbl 1166.60025
[12] Pitman, J., One-dimensional Brownian motion and the three-dimensional Bessel process, Adv. Appl. Probab., 7, 511-526 (1975) · Zbl 0332.60055
[13] Sasirekha, R.; Rakkiyappan, R.; Cao, J.; Wan, Y.; Alsaedi, A., \(H_\infty\) state estimation of discrete-time Markov jump neural networks with general transition probabilities and output quantization, J. Difference Equ. Appl., 23, 11, 1824-1852 (2017) · Zbl 1383.37075
[14] Sato, K., Criteria of weak and strong transience for Lévy processes, (Proceedings of Japan-Russia Symposium on Probability Theory and Mathematical Statistics (1995), World Scientific Press), 438-449 · Zbl 0963.60043
[15] Sato, K.; Watanabe, T.; Yamamuro, K.; Yamazato, M., Multidimensional process of Ornstein-Uhlenbeck type with nondiagonalizable matrix in linear drift terms, Nagoya Math. J., 141, 45-78 (1996) · Zbl 0938.60039
[16] Sato, K.; Watanabe, T.; Yamazato, M., Recurrence conditions for multidimensional processes of Ornstein-Uhlenbeck type, J. Math. Soc. Japan, 46, 2, 245-265 (1994) · Zbl 0815.60071
[17] Shiga, T., A recurrence criterion for Markov processes of Ornstein-Uhlenbeck type, Probab. Theory Related Fields, 85, 425-447 (1990) · Zbl 0677.60088
[18] Syed Ali, M.; Yogambigai, J.; Cao, J., Synchronization of master-slave Markovian switching complex dynamical networks with time-varying delays in nonlinear function via sliding mode control, Acta Math. Sci., 37, 2, 368-384 (2017) · Zbl 1389.93241
[19] Takeuchi, J., Moments of the last exit times, Proc. Japan Acad., 43, 355-360 (1967) · Zbl 0178.19403
[20] Wang, Z., The joint distributions of first hitting and last exit for Brownian motion, Chinese Sci. Bull., 40, 451-457 (1995) · Zbl 0824.60086
[21] Wang, Z., Some joint distributions for Markov processes, Chinese Sci. Bull., 41, 1321-1327 (1996) · Zbl 0872.60058
[22] Xiao, Y., Asymptotic results for self-similar Markov processes, (Asymptotic Methods in Probability and Statistics (1998)), 323-340 · Zbl 0936.60060
[23] Yamamuro, K., On transient Markov processes of Ornstein-Uhlenbeck type, Nagoya Math. J., 149, 19-32 (1998) · Zbl 0908.60074
[24] Zhu, Q.; Cao, J., Stability of Markovian jump neural networks with impulse control and time varying delays, Nonlinear Anal. RWA, 13, 5, 2259-2270 (2012) · Zbl 1254.93157
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