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Lagging and leading coupled continuous time random walks, renewal times and their joint limits

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  • Straka, P.
  • Henry, B.I.
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    Abstract

    Subordinating a random walk to a renewal process yields a continuous time random walk (CTRW), which models diffusion and anomalous diffusion. Transition densities of scaling limits of power law CTRWs have been shown to solve fractional Fokker-Planck equations. We consider limits of CTRWs which arise when both waiting times and jumps are taken from an infinitesimal triangular array. Two different limit processes are identified when waiting times precede jumps or follow jumps, respectively, together with two limit processes corresponding to the renewal times. We calculate the joint law of all four limit processes evaluated at a fixed time t.

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    Bibliographic Info

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

    Volume (Year): 121 (2011)
    Issue (Month): 2 (February)
    Pages: 324-336

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    Handle: RePEc:eee:spapps:v:121:y:2011:i:2:p:324-336

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    Keywords: Continuous time random walk Stochastic process limit Lévy process Time-change Subordination Triangular array Renewal times Skorokhod space Subdiffusion;

    References

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    1. Meerschaert, Mark M. & Scheffler, Hans-Peter, 2008. "Triangular array limits for continuous time random walks," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1606-1633, September.
    2. Enrico Scalas & Rudolf Gorenflo & Francesco Mainardi, 2000. "Fractional calculus and continuous-time finance," Papers cond-mat/0001120, arXiv.org.
    3. 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.
    4. Raberto, Marco & Scalas, Enrico & Mainardi, Francesco, 2002. "Waiting-times and returns in high-frequency financial data: an empirical study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 314(1), pages 749-755.
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    Cited by:
    1. 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.
    2. 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.
    3. Scalas, Enrico & Viles, Noèlia, 2014. "A functional limit theorem for stochastic integrals driven by a time-changed symmetric α-stable Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 385-410.

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