Properties of switching jump diffusions: maximum principles and Harnack inequalities. (English) Zbl 1440.60075

This authors derive a maximum principle and Harnack inequalities for a class of jump-diffusion processes with regime switching. In comparison with continuous switching diffusions, the associated operators are non-local and lead to systems of integro-differential equations. Those results should be applicable in characterizing recurrence and ergodicity for switching jump diffusions.


60J60 Diffusion processes
60J76 Jump processes on general state spaces
Full Text: DOI arXiv Euclid


[1] Arapostathis, A., Ghosh, M.K. and Marcus, S.I. (1999). Harnack’s inequality for cooperative weakly coupled elliptic systems. Comm. Partial Differential Equations24 1555–1571. · Zbl 0934.35039
[2] Athreya, S. and Ramachandran, K. (2017). Harnack inequality for non-local Schrödinger operators. Potential Anal. To appear. DOI:10.1007/s11118-017-9646-6.
[3] Bass, R.F. and Kassmann, M. (2005). Harnack inequalities for non-local operators of variable order. Trans. Amer. Math. Soc.357 837–850. · Zbl 1052.60060
[4] Bass, R.F., Kassmann, M. and Kumagai, T. (2010). Symmetric jump processes: Localization, heat kernels and convergence. Ann. Inst. Henri Poincaré B, Probab. Stat.46 59–71. · Zbl 1201.60078
[5] Bass, R.F. and Levin, D.A. (2002). Harnack inequalities for jump processes. Potential Anal.17 375–388. · Zbl 0997.60089
[6] Caffarelli, L. and Silvestre, L. (2009). Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math.62 597–638. · Zbl 1170.45006
[7] Chen, X., Chen, Z.-Q., Tran, K. and Yin, G. (2017). Recurrence and ergodicity for a class of regime-switching jump diffusions. Appl. Math. Optim. To appear. DOI:10.1007/s00245-017-9470-9.
[8] Chen, Z.-Q., Hu, E., Xie, L. and Zhang, X. (2017). Heat kernels for non-symmetric diffusion operators with jumps. J. Differential Equations263 6576–6634. · Zbl 1386.35131
[9] Chen, Z.-Q. and Kumagai, T. (2003). Heat kernel estimates for stable-like processes on \(d\)-sets. Stochastic Process. Appl.108 27–62. · Zbl 1075.60556
[10] Chen, Z.-Q. and Kumagai, T. (2008). Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Related Fields140 277–317. · Zbl 1131.60076
[11] Chen, Z.-Q. and Kumagai, T. (2010). A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps. Rev. Mat. Iberoam.26 551–589. · Zbl 1200.60065
[12] Chen, Z.-Q., Wang, H. and Xiong, J. (2012). Interacting superprocesses with discontinuous spatial motion. Forum Math.24 1183–1223. · Zbl 1255.60149
[13] Chen, Z.-Q. and Zhao, Z. (1996). Potential theory for elliptic systems. Ann. Probab.24 293–319. · Zbl 0854.60062
[14] Chen, Z.-Q. and Zhao, Z. (1997). Harnack principle for weakly coupled elliptic systems. J. Differential Equations139 261–282. · Zbl 0882.35039
[15] Evans, L.C. (2010). Partial Differential Equations, 2nd ed. Providence, RI: Amer. Math. Soc. · Zbl 1194.35001
[16] Foondun, M. (2009). Harmonic functions for a class of integro-differential operators. Potential Anal.31 21–44. · Zbl 1171.60018
[17] Ikeda, N., Nagasawa, M. and Watanabe, S. (1966). A construction of Markov process by piecing out. Proc. Jpn. Acad.42 370–375. · Zbl 0178.53401
[18] Jasso-Fuentes, H. and Yin, G. (2013). Advanced Criteria for Controlled Markov-Modulated Diffusions in an Infinite Horizon: Overtaking, Bias, and Blackwell Optimality. Beijing: Science Press.
[19] Komatsu, T. (1973). Markov processes associated with certain integro-differential. Osaka J. Math.10 271–303. · Zbl 0284.60066
[20] Krylov, N.V. (1987). Nonlinear Elliptic and Parabolic Equations of the Second Order. Mathematics and Its Applications (Soviet Series) 7. Dordrecht: D. Reidel Publishing Co. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. · Zbl 0619.35004
[21] Kushner, H.J. (1990). Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems. Systems & Control: Foundations & Applications3. Boston, MA: Birkhäuser, Inc.
[22] Liu, R. (2016). Optimal stopping of switching diffusions with state dependent switching rates. Stochastics88 586–605. · Zbl 1337.60075
[23] Mao, X. and Yuan, C. (2006). Stochastic Differential Equations with Markovian Switching. London: Imperial College Press. · Zbl 1126.60002
[24] Meyer, P. (1975). Renaissance, recollements, mélanges, relentissement de processus de Markov. Ann. Inst. Fourier (Grenoble) 25 465–497. · Zbl 0304.60041
[25] Mikulevičius, R. and Pragarauskas, H. (1988). On Hölder continuity of solutions of certain integro-differential equations. Ann. Acad. Sci. Fenn., Ser. A 1 Math.13 231–238.
[26] Negoro, A. and Tsuchiya, M. (1989). Stochastic processes and semigroups associate with degenerate Lévy generating operators. Stoch. Stoch. Rep.26 29–61. · Zbl 0678.60063
[27] Protter, M.H. and Weinberger, H.F. (1967). Maximum Principles in Differential Equations. Englewood Cliffs, NJ: Prentice-Hall, Inc. · Zbl 0153.13602
[28] Sharpe, M. (1986). General Theory of Markov Processes. New York: Academic. · Zbl 0649.60079
[29] Song, R. and Vondraček, Z. (2005). Harnack inequality for some discontinuous Markov processes with a diffusion part. Glas. Mat. Ser. III40 177–187.
[30] Wang, J.-M. (2014). Martingale problems for switched processes. Math. Nachr.287 1186–1201. · Zbl 1314.60141
[31] Xi, F. (2009). Asymptotic properties of jump-diffusion processes with state-dependent switching. Stoch. Process. Appl.119 2198–2221. · Zbl 1191.60091
[32] Yin, G. and Zhu, C. (2010). Hybrid Switching Diffusions: Properties and Applications. · Zbl 1279.60007
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