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Triangular array limits for continuous time random walks


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  • Meerschaert, Mark M.
  • Scheffler, Hans-Peter
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    A continuous time random walk (CTRW) is a random walk subordinated to a renewal process, used in physics to model anomalous diffusion. Transition densities of CTRW scaling limits solve fractional diffusion equations. This paper develops more general limit theorems, based on triangular arrays, for sequences of CTRW processes. The array elements consist of random vectors that incorporate both the random walk jump variable and the waiting time preceding that jump. The CTRW limit process consists of a vector-valued Lévy process whose time parameter is replaced by the hitting time process of a real-valued nondecreasing Lévy process (subordinator). We provide a formula for the distribution of the CTRW limit process and show that their densities solve abstract space-time diffusion equations. Applications to finance are discussed, and a density formula for the hitting time of any strictly increasing subordinator is developed.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 118 (2008)
    Issue (Month): 9 (September)
    Pages: 1606-1633

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    Handle: RePEc:eee:spapps:v:118:y:2008:i:9:p:1606-1633

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    Keywords: Continuous time random walk Subordinator Hitting time Fractional Cauchy problem;


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    1. Benoit Mandelbrot, 1963. "The Variation of Certain Speculative Prices," The Journal of Business, University of Chicago Press, vol. 36, pages 394.
    2. Meerschaert, Mark M. & Scheffler, Hans-Peter, 2006. "Stochastic model for ultraslow diffusion," Stochastic Processes and their Applications, Elsevier, vol. 116(9), pages 1215-1235, September.
    3. Francesco Mainardi & Marco Raberto & Rudolf Gorenflo & Enrico Scalas, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Papers cond-mat/0006454,, revised Nov 2000.
    4. Scalas, Enrico & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
    5. Marco Raberto & Enrico Scalas & Francesco Mainardi, 2004. "Waiting-times and returns in high-frequency financial data: an empirical study," Finance 0411014, EconWPA.
    6. Loretan, Mico & Phillips, Peter C. B., 1994. "Testing the covariance stationarity of heavy-tailed time series: An overview of the theory with applications to several financial datasets," Journal of Empirical Finance, Elsevier, vol. 1(2), pages 211-248, January.
    7. Meerschaert, Mark M. & Scalas, Enrico, 2006. "Coupled continuous time random walks in finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(1), pages 114-118.
    8. Bondesson, Lennart & Kristiansen, Gundorph K. & Steutel, Fred W., 1996. "Infinite divisibility of random variables and their integer parts," Statistics & Probability Letters, Elsevier, vol. 28(3), pages 271-278, July.
    9. Baeumer, B. & Benson, D.A. & Meerschaert, M.M., 2005. "Advection and dispersion in time and space," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 350(2), pages 245-262.
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    Cited by:
    1. Meerschaert, Mark M. & Nane, Erkan & Xiao, Yimin, 2013. "Fractal dimension results for continuous time random walks," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1083-1093.
    2. Cohen, Serge & Meerschaert, Mark M. & Rosinski, Jan, 2010. "Modeling and simulation with operator scaling," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2390-2411, December.
    3. D’Ovidio, Mirko, 2012. "From Sturm–Liouville problems to fractional and anomalous diffusions," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3513-3544.
    4. Mijena, Jebessa B. & Nane, Erkan, 2014. "Correlation structure of time-changed Pearson diffusions," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 68-77.
    5. Straka, P. & Henry, B.I., 2011. "Lagging and leading coupled continuous time random walks, renewal times and their joint limits," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 324-336, February.
    6. Barczyk, A. & Kern, P., 2013. "Scaling limits of coupled continuous time random walks and residual order statistics through marked point processes," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 796-812.
    7. Fernandez-Anaya, G. & Valdes-Parada, F.J. & Alvarez-Ramirez, J., 2011. "On generalized fractional Cattaneo’s equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(23), pages 4198-4202.
    8. Kumar, A. & Nane, Erkan & Vellaisamy, P., 2011. "Time-changed Poisson processes," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1899-1910.


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