Advertisement

Advances in Computational Mathematics

, Volume 45, Issue 2, pp 787–811 | Cite as

Fokker-Planck equation driven by asymmetric Lévy motion

  • Xiao Wang
  • Wenpeng Shang
  • Xiaofan LiEmail author
  • Jinqiao Duan
  • Yanghong Huang
Article
  • 39 Downloads

Abstract

Non-Gaussian Lévy noises are present in many models for understanding underlining principles of physics, finance, biology, and more. In this work, we consider the Fokker-Planck equation (FPE) due to one-dimensional asymmetric Lévy motion, which is a non-local partial differential equation. We present an accurate numerical quadrature for the singular integrals in the non-local FPE and develop a fast summation method to reduce the order of the complexity from O(J2) to \(O(J\log J)\) in one time step, where J is the number of unknowns. We also provide conditions under which the numerical schemes satisfy maximum principle. Our numerical method is validated by comparing with exact solutions for special cases. We also discuss the properties of the probability density functions and the effects of various factors on the solutions, including the stability index, the skewness parameter, the drift term, the Gaussian and non-Gaussian noises, and the domain size.

Keywords

Fokker-Planck equations Non-Gaussian noises Asymmetric α-stable Lévy motion Non-local partial differential equation Fast algorithm 

Mathematics Subject Classification (2010)

65M06 35Q84 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Funding information

The research is partially supported by the grants China Scholarship Council no. 201306160071 (X.W.), NSF-DMS no. 1620449 (J.D. and X.L.), and NNSFs of China nos. 11531006 and 11771449 (J.D.).

References

  1. 1.
    Abels, H., Kassmann, M.: The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels. Osaka. J. Math 46(3), 661–683 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Acosta, G., Borthagaray, J.: A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 55(2), 472–495 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Acosta, G., Borthagaray, J., Bruno, O., Maas, M.: Regularity theory and high order numerical methods for the (1D)-fractional Laplacian. Math. Comput. 87, 1821–1857 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Applebaum, D.: Lévy processes and stochastic calculus, 2nd edn. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cartea, A., del Castillo-Negrete, D.: Fractional diffusion models of option prices in markets with jumps. Physica A 374(2), 749–763 (2007)CrossRefGoogle Scholar
  6. 6.
    Chen, Z., Hu, E., Xie, L., Zhang, X.: Heat kernels for non-symmetric diffusion operators with jumps. J. Differ. Equations 263(10), 6576–6634 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, Z., Zhang, X.: Heat kernels and analyticity of non-symmetric jump diffusion semigroups. Probab. Theory Rel. 165(1), 267–312 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cozzi, M.: Interior regularity of solutions of non-local equations in Sobolev and Nikol’skii spaces. Ann. Mat. Pura. Appl. 196(2), 555–578 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D’Elia, M., Gunzburger, M.: The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator. Comput. Math. Appl. 66(7), 1245–1260 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54(4), 667–696 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Duan, J.: An introduction to stochastic dynamics. Cambridge University Press, New York (2015)zbMATHGoogle Scholar
  12. 12.
    Gao, T., Duan, J., Li, X., Song, R.: Mean exit time and escape probability for dynamical systems driven by Lévy noise. SIAM J. Sci. Comput. 36(3), A887–A906 (2014)CrossRefzbMATHGoogle Scholar
  13. 13.
    Gardiner, C.W.: Handbook of stochastic methods, 3rd edn. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  14. 14.
    Golub, G.H., Van Loan, C.F.: Matrix computations, 4th edn. JHU Press, Baltimore (2012)zbMATHGoogle Scholar
  15. 15.
    Grubb, G.: Fractional Laplacians on domains, a development of Hormander’s theory of u-transmission pseudodifferential operators. Adv. Math. 268, 478–528 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hao, M., Duan, J., Song, R., Xu, W.: Asymmetric non-Gaussian effects in a tumor growth model with immunization. Appl. Math. Model. 38, 4428–4444 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hein, C., Imkeller, P., Pavlyukevich, I.: Limit theorems for p-variations of solutions of SDEs driven by additive stable Lévy noise and model selection for paleo-climatic data. Interdiscip. Math. Sci. 8, 137–150 (2009)zbMATHGoogle Scholar
  18. 18.
    Huang, Q., Duan, J., Wu, J.: Maximum principles for nonlocal parabolic Waldenfels operators. Bull. Math. Sci., published online.  https://doi.org/10.1007/s13373-018-0126-0 (2018)
  19. 19.
    Huang, Y., Oberman, A.: Numerical methods for the fractional Laplacian: a finite difference-quadrature approach. SIAM J. Numer. Anal. 52(6), 3056–3084 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Humphries, N.E., Queiroz, N., Dyer, J.R., Pade, N.G., Musyl, M.K., Schaefer, K.M., Fuller, D.W., Brunnschweiler, J.M., Doyle, T.K., Houghton, J.D., et al.: Environmental context explains Lévy and Brownian movement patterns of marine predators. Nature 465(7301), 1066–1069 (2010)CrossRefGoogle Scholar
  21. 21.
    Koren, T., Chechkin, A., Klafter, J.: On the first passage time and leapover properties of Lévy mmotion. Physica A 379, 10–22 (2007)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kwasnicki, M.: Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20(1), 7–51 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 166, 209–219 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mao, Z., Shen, J.: Efficient Spectral-Galerkin method for fractional partial differential equations with variable coefficients. J. Comput. Phys. 307, 243–261 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mao, Z., Shen, J.: Hermite spectral methods for fractional PDEs in unbounded domains. SIAM J. Sci. Comput. 39(5), A1928–A1950 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Riabiz, M., Godsill, S.: Approximate simulation of linear continuous time models driven by asymmetric stable Lévy processes. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), New Orleans, LA, 2017 pp. 4676–4680 (2017)Google Scholar
  27. 27.
    Risken, H.: The Fokker-Planck equation methods of solution and applications, 2nd edn. Springer, Berlin (1996)zbMATHGoogle Scholar
  28. 28.
    Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary. J. Math. Pure. Appl. 101(3), 275–302 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Samorodnitsky, G., Taqqu, M.S.: Stable non-Gaussian random process. Chapman & Hall/CRC (1994)Google Scholar
  30. 30.
    Sato, K. I.: Lévy processes and infinitely divisible distributions. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  31. 31.
    Schertzer, D., Larcheveque, M., Duan, J, Yanovsky, V.V., Lovejoy, S.: Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises. J. Math. Phys. 42(1), 200–212 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Sidi, A., Israeli, M.: Quadrature methods for periodic singular and weakly singular Fredholm integral equtaions. J. Sci. Comput. 3(2), 201–231 (1998)CrossRefzbMATHGoogle Scholar
  33. 33.
    Song, R., Xie, L. arXiv:1806.09033 (2018)
  34. 34.
    Srokowski, T.: Asymmetric Lévy flights in nonhomogeneous environments. J. Stat. Mech.-Theory E. 2014(5), P05,024 (2014)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Tian, X., Du, Q.: Nonconforming discontinuous Galerkin methods for nonlocal variational problems. SIAM J. Numer. Anal. 53(2), 762–781 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Wang, M., Duan, J.: Existence and regularity of a linear nonlocal Fokker-Planck equation with growing drift. J. Math. Anal. Appl. 449(1), 228–243 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wang, X., Duan, J., Li, X., Luan, Y.: Numerical methods for the mean exit time and escape probability of two-dimensional stochastic dynamical systems with non-Gaussian noises. Appl. Math. Comput. 258, 282–295 (2015)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Wang, X., Duan, J., Li, X., Song, R.: Numerical algorithms for mean exit time and escape probability of stochastic systems with asymmetric Lévy motion. Appl. Math. Comput. 337, 618–634 (2018)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Wei, J., Tian, R.: Well-posedness for the fractional Fokker-Planck equations. J. Math. Phys. 56(3), 1–12 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Xu, Y., Feng, J., Li, J., Zhang, H.: Lévy noise induced switch in the gene transcriptional regulatory system. Chaos 23(1), 013,110 (2013)CrossRefGoogle Scholar
  41. 41.
    Zeng, L., Xu, B.: Effects of asymmetric Lévy noise in parameter-induced aperiodic stochastic resonance. Physica A 389, 5128–5136 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHenan UniversityKaifengChina
  2. 2.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA
  3. 3.School of MathematicsUniversity of ManchesterManchesterUK

Personalised recommendations