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Extremal behavior of a coupled continuous time random walk

Author

Listed:
  • Schumer, Rina
  • Baeumer, Boris
  • Meerschaert, Mark M.

Abstract

Coupled continuous time random walks (CTRWs) model normal and anomalous diffusion of random walkers by taking the sum of random jump lengths dependent on the random waiting times immediately preceding each jump. They are used to simulate diffusion-like processes in econophysics such as stock market fluctuations, where jumps represent financial market microstructure like log returns. In this and many other applications, the magnitude of the largest observations (e.g. a stock market crash) is of considerable importance in quantifying risk. We use a stochastic process called a coupled continuous time random maxima (CTRM) to determine the density governing the maximum jump length of a particle undergoing a CTRW. CTRM are similar to continuous time random walks but track maxima instead of sums. The many ways in which observations can depend on waiting times can produce an equally large number of CTRM governing density shapes. We compare densities governing coupled CTRM with their uncoupled counterparts for three simple observation/wait dependence structures.

Suggested Citation

  • Schumer, Rina & Baeumer, Boris & Meerschaert, Mark M., 2011. "Extremal behavior of a coupled continuous time random walk," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(3), pages 505-511.
  • Handle: RePEc:eee:phsmap:v:390:y:2011:i:3:p:505-511
    DOI: 10.1016/j.physa.2010.10.018
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    References listed on IDEAS

    as
    1. 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.
    2. Masoliver, Jaume & Montero, Miquel & Perelló, Josep & Weiss, George H., 2007. "The CTRW in finance: Direct and inverse problems with some generalizations and extensions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(1), pages 151-167.
    3. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
    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. Broszkiewicz-Suwaj, Ewa & Jurlewicz, Agnieszka, 2008. "Pricing on electricity market based on coupled-continuous-time-random-walk concept," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(22), pages 5503-5510.
    6. Pancheva, Elisaveta & Mitov, Ivan K. & Mitov, Kosto V., 2009. "Limit theorems for extremal processes generated by a point process with correlated time and space components," Statistics & Probability Letters, Elsevier, vol. 79(3), pages 390-395, February.
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    9. 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.
    10. Guido Germano & Mauro Politi & Enrico Scalas & Ren'e L. Schilling, 2008. "Stochastic calculus for uncoupled continuous-time random walks," Papers 0802.3769, arXiv.org, revised Jan 2009.
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