Markov Chain Approximation of Pure Jump Processes

  • Ante Mimica
  • Nikola Sandrić
  • René L. Schilling


In this paper we discuss weak convergence of continuous-time Markov chains to a non-symmetric pure jump process. We approach this problem using Dirichlet forms as well as semimartingales. As an application, we discuss how to approximate a given Markov process by Markov chains.


Non-symmetric Dirichlet form Non-symmetric Hunt process Markov chain Mosco convergence Semimartingale Semimartingale characteristics Weak convergence 

Mathematics Subject Classification (2010)

60J25 60J27 60J75 



Financial support through the Croatian Science Foundation (under Project 3526) (for Ante Mimica), the Croatian Science Foundation (under Project 3526) and the Dresden Fellowship Programme (for Nikola Sandrić) is gratefully acknowledged.


  1. 1.
    Bass, R.F.: Uniqueness in law for pure jump Markov processes. Probab. Theory Relat. Fields 79(2), 271–287 (1988) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Burdzy, K., Chen, Z.-Q.: Discrete approximations to reflected Brownian motion. Ann. Probab. 36(2), 698–727 (2008) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bass, R.F., Kumagai, T.: Symmetric Markov chains on \(\mathbb{Z}^{d}\) with unbounded range. Trans. Am. Math. Soc. 360(4), 2041–2075 (2008) CrossRefMATHGoogle Scholar
  4. 4.
    Bass, R.F., Kassmann, M., Kumagai, T.: Symmetric jump processes: localization, heat kernels and convergence. Ann. Inst. Henri Poincaré Probab. Stat. 46(1), 59–71 (2010) MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bass, R.F., Kumagai, T., Uemura, T.: Convergence of symmetric Markov chains on \(\mathbb{Z}^{d}\). Probab. Theory Relat. Fields 148(1–2), 107–140 (2010) CrossRefMATHGoogle Scholar
  6. 6.
    Böttcher, B., Schilling, R.L., Wang, J.: Lévy Matters. III. Springer, Cham (2013) CrossRefMATHGoogle Scholar
  7. 7.
    Chen, Z.-Q., Kim, P., Kumagai, T.: Discrete approximation of symmetric jump processes on metric measure spaces. Probab. Theory Relat. Fields 155(3–4), 703–749 (2013) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Deuschel, J-D., Kumagai, T.: Markov chain approximations to nonsymmetric diffusions with bounded coefficients. Commun. Pure Appl. Math. 66(6), 821–866 (2013) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ethier, S.N., Kurtz, T.G.: Markov Processes. John Wiley & Sons Inc., New York (1986) CrossRefMATHGoogle Scholar
  10. 10.
    Folland, G.B.: Real Analysis. John Wiley & Sons, Inc., New York (1984) MATHGoogle Scholar
  11. 11.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter & Co., Berlin (2011) MATHGoogle Scholar
  12. 12.
    Fukushima, M., Uemura, T.: Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms. Ann. Probab. 40(2), 858–889 (2012) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hino, M.: Convergence of non-symmetric forms. J. Math. Kyoto Univ. 38(2), 329–341 (1998) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Husseini, R., Kassmann, M.: Markov chain approximations for symmetric jump processes. Potential Anal. 27(4), 353–380 (2007) MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hoh, W.: Pseudo differential operators with negative definite symbols of variable order. Rev. Mat. Iberoam. 16(2), 219–241 (2000) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Jacob, N.: Pseudo Differential Operators and Markov Processes, vol. III. Imperial College Press, London (2005) CrossRefMATHGoogle Scholar
  17. 17.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003) CrossRefMATHGoogle Scholar
  18. 18.
    Kim, P.: Weak convergence of censored and reflected stable processes. Stoch. Process. Appl. 116(12), 1792–1814 (2006) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kuwae, K., Shioya, T.: Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry. Commun. Anal. Geom. 11(4), 599–673 (2003) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kühn, F.: Probability and heat kernel estimates for Lévy(-type) Processes. PhD thesis, der Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden (2016) Google Scholar
  21. 21.
    Kühn, F.: Solutions of Lévy-driven SDEs with unbounded coefficients as Feller processes. Preprint (2016). Available at arXiv:1610.02286
  22. 22.
    Mosco, U.: Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123(2), 368–421 (1994) MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Ma, Z.M., Röckner, M.: Introduction to the Theory of (Nonsymmetric) Dirichlet Forms. Springer, Berlin (1992) CrossRefMATHGoogle Scholar
  24. 24.
    Negoro, A.: Stable-like processes: construction of the transition density and the behavior of sample paths near \(t=0\). Osaka J. Math. 31(1), 189–214 (1994) MathSciNetMATHGoogle Scholar
  25. 25.
    Norris, J.R.: Markov Chains. Cambridge University Press, Cambridge (1998) MATHGoogle Scholar
  26. 26.
    Schilling, R.L.: Growth and Hölder conditions for the sample paths of Feller processes. Probab. Theory Relat. Fields 112(4), 565–611 (1998) CrossRefMATHGoogle Scholar
  27. 27.
    Schilling, R.L., Schnurr, A.: The symbol associated with the solution of a stochastic differential equation. Electron. J. Probab. 15, 1369–1393 (2010) MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Schilling, R.L., Uemura, T.: On the structure of the domain of a symmetric jump-type Dirichlet form. Publ. Res. Inst. Math. Sci. 48(1), 1–20 (2012) MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (2006). Reprint of the 1997 edition MATHGoogle Scholar
  30. 30.
    Schilling, R.L., Wang, J.: Some theorems on Feller processes: transience, local times and ultracontractivity. Trans. Am. Math. Soc. 365(6), 3255–3268 (2013) MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Schilling, R.L., Wang, J.: Lower bounded semi-Dirichlet forms associated with Lévy type operators. In: Chen, Z-Q., Jacob, N., Takeda, M., Uemura, T. (eds.) Festschrift Masatoshi Fukushima. In Honor of Masatoshi Fukushima’s Sanju, pp. 507–527. World Scientific, New Jersey (2015) CrossRefGoogle Scholar
  32. 32.
    Stroock, D.W., Zheng, W.: Markov chain approximations to symmetric diffusions. Ann. Inst. Henri Poincaré Probab. Stat. 33(5), 619–649 (1997) MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Tölle, J.M.: Convergence of non-symmetric forms with changing reference measures. Master’s thesis, Faculty of Mathematics, University of Bielefeld (2006) Google Scholar

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Authors and Affiliations

  • Ante Mimica
  • Nikola Sandrić
    • 1
  • René L. Schilling
    • 2
  1. 1.Faculty of Civil EngineeringUniversity of ZagrebZagrebCroatia
  2. 2.Institut für Mathematische Stochastik, Fachrichtung MathematikTU DresdenDresdenGermany

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