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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.

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.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:2:p:324-336
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    4. 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.
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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. Vassili N. Kolokoltsov, 2023. "Fractional Equations for the Scaling Limits of Lévy Walks with Position-Dependent Jump Distributions," Mathematics, MDPI, vol. 11(11), pages 1-19, June.
    3. 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.
    4. 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.
    5. Busani, Ofer, 2017. "Finite dimensional Fokker–Planck equations for continuous time random walk limits," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1496-1516.
    6. Straka, Peter, 2018. "Variable order fractional Fokker–Planck equations derived from Continuous Time Random Walks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 503(C), pages 451-463.
    7. Buraczewski, Dariusz & Dyszewski, Piotr & Iksanov, Alexander & Marynych, Alexander, 2020. "Random walks in a strongly sparse random environment," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 3990-4027.
    8. Peggy Cénac & Arnaud Ny & Basile Loynes & Yoann Offret, 2019. "Persistent Random Walks. II. Functional Scaling Limits," Journal of Theoretical Probability, Springer, vol. 32(2), pages 633-658, June.
    9. Beghin, Luisa & Macci, Claudio & Ricciuti, Costantino, 2020. "Random time-change with inverses of multivariate subordinators: Governing equations and fractional dynamics," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 6364-6387.
    10. Kelbert, M. & Konakov, V. & Menozzi, S., 2016. "Weak error for Continuous Time Markov Chains related to fractional in time P(I)DEs," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 1145-1183.
    11. Magdziarz, M. & Scheffler, H.P. & Straka, P. & Zebrowski, P., 2015. "Limit theorems and governing equations for Lévy walks," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4021-4038.

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