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Stochastic calculus for uncoupled continuous-time random walks

Author

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  • Guido Germano
  • Mauro Politi
  • Enrico Scalas
  • Ren'e L. Schilling

Abstract

The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications in physics, but also in insurance, finance and economics. A definition is given for a class of stochastic integrals driven by a CTRW, that includes the Ito and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Ito integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral and its Ito integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Levy alpha-stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, that generalize the standard diffusion equation solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE, and check it by Monte Carlo.

Suggested Citation

  • 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.
  • Handle: RePEc:arx:papers:0802.3769
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    Cited by:

    1. 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.
    2. Álvaro Cartea, 2013. "Derivatives pricing with marked point processes using tick-by-tick data," Quantitative Finance, Taylor & Francis Journals, vol. 13(1), pages 111-123, January.
    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.
    4. Dexter Cahoy, 2012. "Moment estimators for the two-parameter M-Wright distribution," Computational Statistics, Springer, vol. 27(3), pages 487-497, September.
    5. Guoxing Lin, 2018. "Analysis of PFG Anomalous Diffusion via Real-Space and Phase-Space Approaches," Mathematics, MDPI, vol. 6(2), pages 1-16, January.
    6. 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.
    7. Chen, Zhen-Qing & Kim, Kyeong-Hun & Kim, Panki, 2015. "Fractional time stochastic partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1470-1499.
    8. Frank Marten & Krasimira Tsaneva-Atanasova & Luca Giuggioli, 2012. "Bacterial Secretion and the Role of Diffusive and Subdiffusive First Passage Processes," PLOS ONE, Public Library of Science, vol. 7(8), pages 1-12, August.

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