Electronic Communications in Probability

On time-changed Gaussian processes and their associated Fokker-Planck-Kolmogorov equations

Marjorie Hahn, Jelena Ryvkina, Kei Kobayashi, and Sabir Umarov

Full-text: Open access

Abstract

This paper establishes Fokker-Planck-Kolmogorov type equations for time-changed Gaussian processes. Examples include those equations for a time-changed fractional Brownian motion with time-dependent Hurst parameter and for a time-changed Ornstein-Uhlenbeck process. The time-change process considered is the inverse of either a stable subordinator or a mixture of independent stable subordinators.

Article information

Source
Electron. Commun. Probab., Volume 16 (2011), paper no. 15, 150-164.

Dates
Accepted: 17 March 2011
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465261971

Digital Object Identifier
doi:10.1214/ECP.v16-1620

Mathematical Reviews number (MathSciNet)
MR2783336

Zentralblatt MATH identifier
1231.60029

Subjects
Primary: 60G15: Gaussian processes
Secondary: 35Q84: Fokker-Planck equations 60G22: Fractional processes, including fractional Brownian motion

Keywords
time-change inverse subordinator Gaussian process Fokker-Planck equation Kolmogorov equation fractional Brownian motion time-dependent Hurst parameter Volterra process

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Hahn, Marjorie; Ryvkina, Jelena; Kobayashi, Kei; Umarov, Sabir. On time-changed Gaussian processes and their associated Fokker-Planck-Kolmogorov equations. Electron. Commun. Probab. 16 (2011), paper no. 15, 150--164. doi:10.1214/ECP.v16-1620. https://projecteuclid.org/euclid.ecp/1465261971


Export citation

References

  • Alòs, E., Mazet, O., Nualart, D. Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 (2001), no. 2, 766-801.
  • Andries, E., Umarov, S., Steinberg, S. Monte Carlo random walk simulations based on distributed order differential equations with applications to cell biology. Fract. Calc. Appl. Anal. 9 (2006), no. 4, 351-369.
  • Applebaum, D. Lévy Processes and Stochastic Calculus, 2nd edition. Cambridge University Press (2009).
  • Baeumer, B., Meerschaert, M. M., Nane, E. Brownian subordinators and fractional Cauchy problems. Trans. Amer. Math. Soc. 361 (2009), no. 7, 3915-3930.
  • Benson, D. A., Wheatcraft, S. W., Meerschaert, M. M. Application of a fractional advection-dispersion equation. Water Resour. Res. 36 (2000), no. 6, 1403-1412.
  • Biagini, F., Hu, Y., Øksendal, B., Zhang, T. Stochastic calculus for fractional Brownian motion and applications. Springer (2008).
  • Cheridito, P. Mixed fractional Brownian motion. Bernoulli. 7 (2001), no. 6, 913-934.
  • Cheridito, P. Arbitrage in fractional Brownian motion models. Finance Stoch. 7 (2003), no. 4, 533-553.
  • Decreusefond, L. Stochastic integration with respect to Volterra processes. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 2, 123-149..
  • Fannjiang, A., Komorowski, T. Fractional Brownian motions and enhanced diffusion in a unidirectional wave-like turbulence. J. Statist. Phys. 100 (2000), no. 5-6, 1071-1095.
  • Friedman, A. Partial Differential Equations of Parabolic Type. Prentice-Hall (1964).
  • Gorenflo, R., Mainardi, F. Fractional calculus: integral and differential equations of fractional order. Fractals and Fractional Calculus in Continuum Mechanics. 223-276, Springer (1997).
  • Gorenflo, R., Mainardi, F. Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal. 1 (1998), no. 2, 167-191.
  • Gorenflo, R., Mainardi, F., Scalas, E., Raberto, M. Fractional calculus and continuous-time finance. III. The diffusion limit. Mathematical Finance. 171-180, Trends Math., Birkháuser, Basel (2001).
  • Hahn, M., Kobayashi, K., Umarov, S. Lévy process and their associated time-fractional order pseudo-differential equations. J. Theoret. Probab. DOI: 10.1007/s10959-010-0289-4 (2010).
  • Hahn, M., Kobayashi, K., Umarov, S. Fokker-Planck-Kolmogorov equations associated with time-changed fractional Brownian motion. Proc. Amer. Math. Soc. 139 (2011), no. 2, 691-705.
  • Hörmander, L. Random walk models for space-fractional diffusion processes. The Analysis of Linear Partial Differential Operators. III. Pseudo-differential operators, 2nd edition, Springer (2007).
  • Janczura, J., Wyłomańska, A. Subdynamics of financial data from fractional Fokker-Planck equation. Acta Phys. Pol. B. 40 (2009), 1341-1351.
  • Janson, S. Gaussian Hilbert Spaces. Cambridge University Press (1997).
  • Kobayashi, K. Stochastic calculus for a time-changed semimartingale and the associated stochastic differential equations. J. Theoret. Probab. DOI: 10.1007/s10959-010-0320-9 (2010).
  • Kochubei, A. N. Distributed order calculus and equations of ultraslow diffusion. J. Math. Anal. Appl. 340 (2008), no. 1, 252-281.
  • Malliavin, P. Stochastic Analysis. Springer Verlag (1997).
  • Meerschaert, M. M., Scheffler, H.-P. Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 41 (2004), no. 3, 623-638.
  • Meerschaert, M. M., Scheffler, H.-P. Stochastic model for ultraslow diffusion. Stochastic Process. Appl. 116 (2006), no. 9, 1215-1235.
  • Meerschaert, M. M., Scheffler, H.-P. Triangular array limits for continuous time random walks. Stochastic Process. Appl. 118 (2008), no. 9, 1606-1633.
  • Metzler, R., Klafter, J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), no. 1, 1-77.
  • Miao, Y., Ren, W., Ren, Z. On the fractional mixed fractional Brownian motion. Appl. Math. Sci. 35 (2008), no. 33-36, 1729-1738.
  • Nualart, D. The Malliavin calculus and related topics, 2nd edition. Springer (2006).
  • Rasmussen, C. E., Williams, C. K. I. Gaussian Processes for Machine Learning. MIT Press (2006).
  • Sato, K-i. Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press (1999).
  • Saxton, M. J., Jacobson, K. Single-particle tracking: applications to membrane dynamics. Annu. Rev. Biophys. Biomol. Struct. 26 (1997), 373-399.
  • Silbergleit, V. M., Gigola, S. V., D'Attellis C. E. Statistical studies of sunspots. Acta Geodaetica et Geophysica Hungarica. 49 (2007), no. 3, 278-283.
  • Thäle, C. Further remarks on mixed fractional Brownian motion. Appl. Math. Sci. 3 (2009), no. 38, 1885-1901.
  • Umarov, S., Gorenflo, R. Cauchy and nonlocal multi-point problems for distributed order pseudo-differential equations. I. Z. Anal. Anwendungen. 24 (2005), no. 3, 449-466.
  • Umarov, S., Gorenflo, R. On multi-dimensional random walk models approximating symmetric space-fractional diffusion processes. Fract. Calc. Appl. Anal. 8 (2005), no. 1, 73-88.
  • Umarov, S., Steinberg, S. Random walk models associated with distributed fractional order differential equations. High Dimensional Probability. 117-127, IMS Lecture Notes Monogr. Ser., 51 (2006).
  • Umarov, S., Steinberg, S. Variable order differential equations with piecewise constant order-function and diffusion with changing modes. Z. Anal. Anwend. 28 (2009), no. 4, 431-450.
  • Zaslavsky, G. M. Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371 (2002), no. 6, 461-580.