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Extremal behavior of a coupled continuous time random walk

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

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    Bibliographic Info

    Article provided by Elsevier in its journal Physica A: Statistical Mechanics and its Applications.

    Volume (Year): 390 (2011)
    Issue (Month): 3 ()
    Pages: 505-511

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    Handle: RePEc:eee:phsmap:v:390:y:2011:i:3:p:505-511

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    Web page: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/

    Related research

    Keywords: Continuous time random walks; Extreme value theory; Power laws; Econophysics;

    References

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    1. Enrico Scalas & Rudolf Gorenflo & Francesco Mainardi, 2000. "Fractional calculus and continuous-time finance," Papers cond-mat/0001120, arXiv.org.
    2. Jaume Masoliver & Miquel Montero & Josep Perello, . "The continuous time random walk formalism in financial markets," Modeling, Computing, and Mastering Complexity 2003 24, Society for Computational Economics.
    3. Jaume Masoliver & Miquel Montero & Josep Perello & George H. Weiss, 2003. "The CTRW in finance: Direct and inverse problems with some generalizations and extensions," Papers cond-mat/0308017, arXiv.org, revised Nov 2006.
    4. Francesco Mainardi & Marco Raberto & Rudolf Gorenflo & Enrico Scalas, 2004. "Fractional calculus and continuous-time finance II: the waiting- time distribution," Finance 0411008, EconWPA.
    5. Mark M. Meerschaert & Enrico Scalas, 2006. "Coupled continuous time random walks in finance," Papers physics/0608281, arXiv.org.
    6. 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.
    7. Jaume Masoliver & Miquel Montero & George H. Weiss, 2002. "A continuous time random walk model for financial distributions," Papers cond-mat/0210513, arXiv.org.
    8. 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.
    9. 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.
    10. 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.
    11. Repetowicz, Przemysław & Richmond, Peter, 2004. "Modeling of waiting times and price changes in currency exchange data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 343(C), pages 677-693.
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