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Fractional order Fokker-Planck-Kolmogorov equations and associated stochastic processes

  • Sabir Umarov
Part of the Developments in Mathematics book series (DEVM, volume 41)

Abstract

This chapter discusses the connection between pseudo-differential and fractional order differential equations considered in Chapters 2–6 with some random (stochastic) processes defined by stochastic differential equations. We assume that the reader is familiar with basic notions of probability theory and stochastic processes, such as a random variable, its density function, mathematical expectation, characteristic function, etc. Since we are interested only in applications of fractional order ΨDOSS, we do not discuss in detail facts on random processes that are already established and presented in other sources. For details of such notations and related facts we refer the reader to the book by Applebaum [App09] (or [IW81, Sat99]). We only mention some basic notations directly related to our discussions on fractional Fokker-Planck-Kolmogorov equations.

References

  1. [App09]
    Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press (2009)Google Scholar
  2. [BMR01]
    Barndorff-Nielsen, O.E., Mikosch, T., Resnick S. (eds): Lévy processes: Theory and applications. Birkhäuser (2001)Google Scholar
  3. [BC07]
    Baudoin, F., Coutin, L.: Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stoch. Process. Appl. 117 (5), 550–574 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [Baz98]
    Bazhlekova E.: The abstract Cauchy problem for the fractional evolution equation. Frac. Calc. Appl. Anal., 1, 255–270 (1998)MathSciNetzbMATHGoogle Scholar
  5. [Ben03]
    Bender, C.: An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter. Stoch. Process. Appl., 104 (1), 81–106 (2003)CrossRefzbMATHGoogle Scholar
  6. [BGR90]
    Bensoussan A., Glowinski, R., Rascanu, R.: Approximations of Zakai equation by the splitting up method. SIAM Journal on Control and Optimization, 28 (6), 1420–1431 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [Ber96]
    Bertoin, J.: Lévy processes. Cambridge University Press (1996)Google Scholar
  8. [BHOZ08]
    Biagini, F., Hu, Y., Oksendal, B., Zhang, T.: Stochastic calculus for fractional Brownian motion and applications. Springer (2008)Google Scholar
  9. [Bo55]
    Bochner, S.: Harmonic Analysis and the Theory of Probability. California Monographs in Mathematical Science, University of California Press, Berkeley (1955)zbMATHGoogle Scholar
  10. [BK96]
    Budhiraja, A., Kallianpur, G.: Approximation to the solutions of Zakai equations using multiple Wiener and Stratonovich expansions. Stochastics, 56, 271–315 (1966)MathSciNetGoogle Scholar
  11. [Co65]
    Courrége, Ph.: Sur la forme intégro-différentielle des opérateurs de C k dans C satisfaisant au principe du maximum. Sém. Théorie du Potentiel, Exposé 2 (1965/66)Google Scholar
  12. [DPZ02]
    Da Prato, G., Zabczyk, J.: Second order partial differential equations in Hilbert spaces. Cambridge University Press (2002)Google Scholar
  13. [Da87]
    Daum, F.E.: Solution of the Zakai equation by separation of variables. IEEE Transactions on automatic control. AC-32 (10), 941–943 (1987)CrossRefGoogle Scholar
  14. [DU98]
    Decreusefond, L., Üstünel, A.S.: Stochastic analysis of the fractional Brownian motion. Potential Analysis. 10 (2), 177–214 (1998)CrossRefGoogle Scholar
  15. [Edi97]
    Edidin, M.: Lipid microdomains in cell surface membranes. Curr. Opin. Struct. Biol., 7, 528–532 (1997)CrossRefGoogle Scholar
  16. [EN99]
    Engel, K.-J., Nagel, R.: One-parameter Semigroups for Linear Evolution Equations. Springer (1999)Google Scholar
  17. [FKK72]
    Fujisaki M., Kallianpur, G., Kunita, H.: Stochastic differential equations for the nonlinear filtering problem. Osaka J. of Mathematics 9(1), 19–40 (1972)MathSciNetzbMATHGoogle Scholar
  18. [FOT94]
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes. De Gruyter Studies in Mathematics, 19, Walter de Gruyter Verlag, Berlin-New-York (1994)Google Scholar
  19. [GW94]
    Ghosh, R.N., Webb, W.W.: Automated detection and tracking of individual and clustered cell surface low density lipoprotein receptor molecules. Biophys. J., 66, 1301–1318 (1994)CrossRefGoogle Scholar
  20. [HU11]
    Hahn, M.G., Umarov, S.R.: Fractional Fokker-Plank-Kolmogorov type equations and their associated stochastic differential equations. Frac. Calc. Appl. Anal., 14 (1), 56–79 (2011)MathSciNetzbMATHGoogle Scholar
  21. [HKU10]
    Hahn, M.G., Kobayashi, K., Umarov, S.R.: SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations. J. Theoret. Prob., 25 (1), 262–279 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [HKU11]
    Hahn, M.G., Kobayashi, K., Umarov, S.R.: Fokker-Planck-Kolmogorov equations associated with time-changed fractional Brownian motion. Proceed. Amer. Math. Soc., 139 (2), 691–705 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [HKRU11]
    Hahn, M.G., Kobayashi, K., Ryvkina, J., Umarov, S.R.: On time-changed Gaussian processes and their associated Fokker-Planck-Kolmogorov equations. Electron. Commun. Probab. 16, 150–164 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [Hoh00]
    Hoh, W.: Pseudo-differential operators with negative definite symbols of variable order. Rev. Mat. Iberoam, 16 (2), 219–241 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [IW81]
    Ikeda, N., Watanabe, Sh.: Stochastic Differential Equations and Diffusion Processes. Amsterdam-Oxford-New York, North-Holland Publishing Co. (1981)zbMATHGoogle Scholar
  26. [IX00]
    Ito, K., Xiong, K.: Gaussian filters for nonlinear filtering problems. IEEE Transactions on Automatic. Control. 1, 45 (5), 910–927 (2000)Google Scholar
  27. [Jac01]
    Jacob, N.: Pseudo-differential Operators and Markov Processes. Vol. I. Fourier Analysis and Semigroups Vol. II. Generators and Their Potential Theory, Vol. III. Markov Processes and Applications. Imperial College Press, London (2001, 2002, 2005)Google Scholar
  28. [JSc02]
    Jacob, J., Schilling, R.L.:. Lévy-type processes and pseudo-differential operators. In Barndorff-Nielsen, O., Mikosch, T., Resnick S. (eds.), Levy Processes: Theory and Applications. Boston, Bikhäser, 139–168 (2001)CrossRefGoogle Scholar
  29. [Jac79]
    Jacod, J.: Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Mathematics, 714, Springer, Berlin (1979)Google Scholar
  30. [KB61]
    Kalman, R.E., Bucy, R.C.: New results in linear filtering and prediction theory. Journal of basic engineering, 83, 95–108 (1961)MathSciNetCrossRefGoogle Scholar
  31. [Kob11]
    Kobayashi, K. Stochastic calculus for a time-changed semimartingale and the associated stochastic differential equations. J. Theoret. Prob., 24 (3), 789–820 (2011)CrossRefzbMATHGoogle Scholar
  32. [Kol40]
    Kolmogorov, A.N.: Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. Dokl. Acad. Sci. URSS, 26 115–118 (1940)Google Scholar
  33. [Ku90]
    Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press (1990)Google Scholar
  34. [Kus67]
    Kushner, H.J.: Dynamical equations for optimal nonlinear filtering. J. Diff. Eq. 3, 179–190 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  35. [LW12]
    Liang, J.-R., Wang, J., Lǔ, L.-J., Hui, G., Qiu, W.-Y., Ren, F.-Y.: Fractional Fokker-Planck equation and Black-Scholes formula in composite-diffusive regime. J. Stat. Phys., 146, 205–216 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  36. [LS02]
    Lipster, R.Sh., Shiryaev, A.N.: Statistics of Random Processes, I, II. Springer, New-York (2002)Google Scholar
  37. [LMR97]
    Lototsky, S., Mikulevicius, R., Rozovskii, R.: Nonlinear filtering revisited: a spectral approach. SIAM J. Control Optimization. 35, 435–461 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  38. [LQR12]
    Lv, L., Qiu, W., Ren, F.: Fractional Fokker-Planck equation with space and time dependent drift and diffusion. J. Stat. Phys. 149, 619–628 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  39. [Mag09]
    Magdziarz, M.: Black-Scholes formula in subdiffusive regime. J. Stat. Phys. 136, 553–564 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  40. [MGZ14]
    Magdziarz, M., Gajda, J., Zorawik, T.: Comment on fractional Fokker-Planck equation with space and time dependent drift and diffusion. J. Stat. Phys. 154, 1241–1250 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  41. [Mai96]
    Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, Solitons and Fractals. 7 (9), 1461–1477 (1996)Google Scholar
  42. [MLP01]
    Mainardi, F., Luchko, Yu., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation. Frac. Calc. Appl. Anal., 4 (2), 153–192 (2001)MathSciNetzbMATHGoogle Scholar
  43. [MVN68]
    Mandelbrot, B.B., Van Ness, J. W.: Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  44. [MBB01]
    Meerschaert, M.M., Benson, D., Bäumer, B.: Operator Lévy motion and multiscaling anomalous diffusion. Phys. Rev. E 63, 021112–021117 (2001)CrossRefGoogle Scholar
  45. [MNX09]
    Meerschaert, M.M., Nane, E., Xiao, Y.: Correlated continuous time random walks. Stat. Probabil. Lett. 79, 1194–1202 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  46. [MBK99]
    Metzler, R., Barkai, E., Klafter, J.: Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach. Phys. Rev. Lett. 82, 3563–3567 (1999)CrossRefGoogle Scholar
  47. [Nua06]
    Nualart, D.: The Malliavin calculus and related topics, 2nd ed. Springer (2006)Google Scholar
  48. [Pro91]
    Protter, P.: Stochastic Integration and Differential Equations. Springer-Verlag, Berlin-New York (1991)Google Scholar
  49. [Roz90]
    Rozovskii, B.L., Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering. Kluwer Academic Publishers, Dordrecht (1990)CrossRefzbMATHGoogle Scholar
  50. [ST94]
    Samorodnitsky, G., Taqqu, M.S. Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance. Chapman & Hall, New York (1994)zbMATHGoogle Scholar
  51. [Sat99]
    Sato, K-i.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press (1999)Google Scholar
  52. [Sax01]
    Saxton, M.J.: Anomalous Subdiffusion in Fluorescence Photobleaching Recovery: A Monte Carlo Study. Biophys. J., 81(4), 2226–2240 (2001)CrossRefGoogle Scholar
  53. [SJ97]
    Saxton, M.J., Jacobson, K.: Single-particle tracking: applications to membrane dynamics. Ann. Rev. Biophys. Biomol. Struct., 26, 373–399 (1997)CrossRefGoogle Scholar
  54. [SLDYL01]
    Schertzer, D., Larchevêque, 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
  55. [Ron05]
    Situ, R: Theory of stochastic differential equations with jumps and applications. Springer (2005)Google Scholar
  56. [SK06]
    Sokolov, I.M., Klafter, J.: Field-induced dispersion in subdiffusion. Phys. Rev. Lett. 97, 140602 (2006)CrossRefGoogle Scholar
  57. [Tai91]
    Taira, K.: Boundary Value Problems and Markov Processes. Lecture notes in Mathematics, 1499. Springer-Verlag, Berlin-Heidelberg- New York-Tokyo (1991)Google Scholar
  58. [Tsa09]
    Tsallis, C.: Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World. Springer, New York (2009)Google Scholar
  59. [UDN14]
    Umarov, S.R., Daum, F., Nelson, K.: Fractional generalizations of filtering problems and their associated fractional Zakai equations. Frac. Calc. and Appl. Anal., 17 (3), 745–764 (2014)MathSciNetGoogle Scholar
  60. [UZ99]
    Uchaykin, V.V., Zolotarev, V.M.: Chance and Stability. Stable Distributions and their Applications. VSP, Utrecht (1999)CrossRefGoogle Scholar
  61. [Wie28]
    Wiener, N.: Differential space. Journal of Mathematical Physics 2, 131–174 (1923)Google Scholar
  62. [WEKN04]
    Weiss, M., Elsner, M., Kartberg, F., Nilsson, T.: Anomalous subdiffusion Is a measure for cytoplasmic crowding in living cells. Biophysical Journal, 87, 3518–3524 (2004)CrossRefGoogle Scholar
  63. [Zak69]
    Zakai, M.: On the optimal filtering of diffusion processes. Z. Wahrsch. Verw. Gebiete. 11 (3), 230–243 (1969)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Sabir Umarov
    • 1
  1. 1.Department of MathematicsUniversity of New HavenWest HavenUSA

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