Markov Jump Processes Approximating a Non-Symmetric Generalized Diffusion

  • Nedžad LimićEmail author


Consider a non-symmetric generalized diffusion X(⋅) in ℝ d determined by the differential operator \(A(\mbox{\boldmath{$x$}})=-\sum_{ij}\partial_{i}a_{ij}(\mbox{\boldmath{$x$}})\partial_{j} +\sum_{i} b_{i}(\mbox{\boldmath{$x$}})\partial_{i}\). In this paper the diffusion process is approximated by Markov jump processes X n (⋅), in homogeneous and isotropic grids G n ⊂ℝ d , which converge in distribution in the Skorokhod space D([0,∞),ℝ d ) to the diffusion X(⋅). The generators of X n (⋅) are constructed explicitly. Due to the homogeneity and isotropy of grids, the proposed method for d≥3 can be applied to processes for which the diffusion tensor \(\{a_{ij}(\mbox{\boldmath{$x$}})\}_{11}^{dd}\) fulfills an additional condition. The proposed construction offers a simple method for simulation of sample paths of non-symmetric generalized diffusion. Simulations are carried out in terms of jump processes X n (⋅). For piece-wise constant functions a ij on ℝ d and piece-wise continuous functions a ij on ℝ2 the construction and principal algorithm are described enabling an easy implementation into a computer code.


Symmetric diffusion Approximation of diffusion Simulation of diffusion Divergence form operators 


  1. 1.
    Bass, R.F., Kumagai, T.: Symmetric Markov chains on ℤd with unbounded range. Trans. Am. Math. Soc. 360, 2041–2075 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Springer, Berlin (1976) zbMATHGoogle Scholar
  3. 3.
    Ethier, S.N., Kurtz, T.G.: Markov Processes, Characteristics and Convergence. Wiley, New York (1986) Google Scholar
  4. 4.
    Étoré, P.: On random walk simulation of one-dimensional diffusion process with discontinuous coefficients. Electron. J. Probab. 11, 249–275 (2006) MathSciNetGoogle Scholar
  5. 5.
    Gikhman, I.I., Skorokhod, A.V.: The Theory of Stochastic Processes, Vol. 1. i 2. Springer, Berlin (1979) Google Scholar
  6. 6.
    Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Partial Differential Equations. Academic Press, New York (1968) Google Scholar
  7. 7.
    Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Am. Math. Soc., Providence (1968) Google Scholar
  8. 8.
    Lejay, A., Martinez, M.: A scheme for simulating one-dimensional diffusion process with discontinuous coefficients. Ann. Appl. Probab. 16(1), 107–139 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Limić, N.: Monte Carlo Simulations of Random Variables, Sequences and Processes. Element, Zagreb (2009) Google Scholar
  10. 10.
    Limić, N., Rogina, M.: Explicit stable methods for second order parabolic systems. Math. Commun. 5, 97–115 (2000) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Limić, N., Rogina, M.: Monotone numerical schemes for a Dirichlet problem for elliptic operators in divergence form. Math. Methods Appl. Sci. 32, 1129–1155 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Maz’ya, V.G.: Sobolev Spaces. Springer, New York (1985) Google Scholar
  13. 13.
    Motzkyn, T.S., Wasov, W.: On the approximation of linear elliptic differential equations by difference equations with positive coefficients. J. Math. Phys. 31, 253–259 (1953) Google Scholar
  14. 14.
    Samarskii, A.A., Matus, P.P., Mazhukin, V.I., Mozolevski, I.E.: Monotone Difference Schemes for Equations with Mixed Derivatives. Comput. Math. Appl. 44, 501–510 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Stein, F.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, New Jersey (1970) zbMATHGoogle Scholar
  16. 16.
    Solonnikov, V.A.: A priori estimates for second order equations of parabolic type. Trudy Math. Inst. Steklov 70, 133–212 (1964) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Solonnikov, V.A.: On a boundary value problem for linear general parabolic system of differential equations. Trudy Math. Inst. Steklov 83 (1965) Google Scholar
  18. 18.
    Stroock, D.W.: Diffusion semigroups corresponding to uniformly elliptic divergence form operators. In: Séminaire de Probabilité XXII. LNMS, vol. 1321, pp. 316–348. Springer, Berlin (1988) CrossRefGoogle Scholar
  19. 19.
    Stroock, D.W., Zheng, W.: Markov chain approximations to symmetric diffusions. Ann. Inst. Henri Poincaré, B Calc. Probab. Stat. 33, 619–649 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Tikhonov, AN, Samarskii, A.A.: On the stability of difference schemes. Dokl. Akad. Nauk SSSR 149, 529–531 (1963) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Dept. of MathematicsUniversity of ZagrebZagrebCroatia

Personalised recommendations