Bernoulli

  • Bernoulli
  • Volume 25, Number 2 (2019), 1045-1075.

Properties of switching jump diffusions: Maximum principles and Harnack inequalities

Xiaoshan Chen, Zhen-Qing Chen, Ky Tran, and George Yin

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Abstract

This work examines a class of switching jump diffusion processes. The main effort is devoted to proving the maximum principle and obtaining the Harnack inequalities. Compared with the diffusions and switching diffusions, the associated operators for switching jump diffusions are non-local, resulting in more difficulty in treating such systems. Our study is carried out by taking into consideration of the interplay of stochastic processes and the associated systems of integro-differential equations.

Article information

Source
Bernoulli, Volume 25, Number 2 (2019), 1045-1075.

Dates
Received: July 2016
Revised: July 2017
First available in Project Euclid: 6 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1551862843

Digital Object Identifier
doi:10.3150/17-BEJ1012

Mathematical Reviews number (MathSciNet)
MR3920365

Zentralblatt MATH identifier
07049399

Keywords
Harnack inequality jump diffusion maximum principle regime switching

Citation

Chen, Xiaoshan; Chen, Zhen-Qing; Tran, Ky; Yin, George. Properties of switching jump diffusions: Maximum principles and Harnack inequalities. Bernoulli 25 (2019), no. 2, 1045--1075. doi:10.3150/17-BEJ1012. https://projecteuclid.org/euclid.bj/1551862843


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References

  • [1] Arapostathis, A., Ghosh, M.K. and Marcus, S.I. (1999). Harnack’s inequality for cooperative weakly coupled elliptic systems. Comm. Partial Differential Equations 24 1555–1571.
  • [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.
  • [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.
  • [5] Bass, R.F. and Levin, D.A. (2002). Harnack inequalities for jump processes. Potential Anal. 17 375–388.
  • [6] Caffarelli, L. and Silvestre, L. (2009). Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62 597–638.
  • [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 Equations 263 6576–6634.
  • [9] Chen, Z.-Q. and Kumagai, T. (2003). Heat kernel estimates for stable-like processes on $d$-sets. Stochastic Process. Appl. 108 27–62.
  • [10] Chen, Z.-Q. and Kumagai, T. (2008). Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Related Fields 140 277–317.
  • [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.
  • [12] Chen, Z.-Q., Wang, H. and Xiong, J. (2012). Interacting superprocesses with discontinuous spatial motion. Forum Math. 24 1183–1223.
  • [13] Chen, Z.-Q. and Zhao, Z. (1996). Potential theory for elliptic systems. Ann. Probab. 24 293–319.
  • [14] Chen, Z.-Q. and Zhao, Z. (1997). Harnack principle for weakly coupled elliptic systems. J. Differential Equations 139 261–282.
  • [15] Evans, L.C. (2010). Partial Differential Equations, 2nd ed. Providence, RI: Amer. Math. Soc.
  • [16] Foondun, M. (2009). Harmonic functions for a class of integro-differential operators. Potential Anal. 31 21–44.
  • [17] Ikeda, N., Nagasawa, M. and Watanabe, S. (1966). A construction of Markov process by piecing out. Proc. Jpn. Acad. 42 370–375.
  • [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.
  • [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].
  • [21] Kushner, H.J. (1990). Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems. Systems & Control: Foundations & Applications 3. Boston, MA: Birkhäuser, Inc.
  • [22] Liu, R. (2016). Optimal stopping of switching diffusions with state dependent switching rates. Stochastics 88 586–605.
  • [23] Mao, X. and Yuan, C. (2006). Stochastic Differential Equations with Markovian Switching. London: Imperial College Press.
  • [24] Meyer, P. (1975). Renaissance, recollements, mélanges, relentissement de processus de Markov. Ann. Inst. Fourier (Grenoble) 25 465–497.
  • [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.
  • [27] Protter, M.H. and Weinberger, H.F. (1967). Maximum Principles in Differential Equations. Englewood Cliffs, NJ: Prentice-Hall, Inc.
  • [28] Sharpe, M. (1986). General Theory of Markov Processes. New York: Academic.
  • [29] Song, R. and Vondraček, Z. (2005). Harnack inequality for some discontinuous Markov processes with a diffusion part. Glas. Mat. Ser. III 40 177–187.
  • [30] Wang, J.-M. (2014). Martingale problems for switched processes. Math. Nachr. 287 1186–1201.
  • [31] Xi, F. (2009). Asymptotic properties of jump-diffusion processes with state-dependent switching. Stoch. Process. Appl. 119 2198–2221.
  • [32] Yin, G. and Zhu, C. (2010). Hybrid Switching Diffusions: Properties and Applications. New York: Springer.