IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i10p1148-d557834.html
   My bibliography  Save this article

Current Trends in Random Walks on Random Lattices

Author

Listed:
  • Jewgeni H. Dshalalow

    (Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32940, USA)

  • Ryan T. White

    (Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32940, USA)

Abstract

In a classical random walk model, a walker moves through a deterministic d -dimensional integer lattice in one step at a time, without drifting in any direction. In a more advanced setting, a walker randomly moves over a randomly configured (non equidistant) lattice jumping a random number of steps. In some further variants, there is a limited access walker’s moves. That is, the walker’s movements are not available in real time. Instead, the observations are limited to some random epochs resulting in a delayed information about the real-time position of the walker, its escape time, and location outside a bounded subset of the real space. In this case we target the virtual first passage (or escape) time. Thus, unlike standard random walk problems, rather than crossing the boundary, we deal with the walker’s escape location arbitrarily distant from the boundary. In this paper, we give a short historical background on random walk, discuss various directions in the development of random walk theory, and survey most of our results obtained in the last 25–30 years, including the very recent ones dated 2020–21. Among different applications of such random walks, we discuss stock markets, stochastic networks, games, and queueing.

Suggested Citation

  • Jewgeni H. Dshalalow & Ryan T. White, 2021. "Current Trends in Random Walks on Random Lattices," Mathematics, MDPI, vol. 9(10), pages 1-38, May.
  • Handle: RePEc:gam:jmathe:v:9:y:2021:i:10:p:1148-:d:557834
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/10/1148/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/9/10/1148/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Scalas, Enrico, 2006. "The application of continuous-time random walks in finance and economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 362(2), pages 225-239.
    2. Takashi Odagaki & Keisuke Kasuya, 2017. "Alzheimer random walk," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 90(9), pages 1-5, September.
    3. Jewgeni H. Dshalalow & Jay Yellen, 1996. "Bulk input queues with quorum and multiple vacations," Mathematical Problems in Engineering, Hindawi, vol. 2, pages 1-12, January.
    4. Jewgeni H. Dshalalow & Ahmed Merie, 2018. "Fluctuation analysis in queues with several operational modes and priority customers," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(2), pages 309-333, July.
    5. Jewgeni H. Dshalalow, 1997. "On the level crossing of multi-dimensional delayed renewal processes," International Journal of Stochastic Analysis, Hindawi, vol. 10, pages 1-7, January.
    6. Abolnikov, Lev M. & Dshalalow, Jewgeni H. & Dukhovny, Alexander M., 1993. "Stochastic analysis of a controlled bulk queueing system with continuously operating server: continuous time parameter queueing process," Statistics & Probability Letters, Elsevier, vol. 16(2), pages 121-128, January.
    7. Ryszard Kutner & Jaume Masoliver, 2017. "The continuous time random walk, still trendy: fifty-year history, state of art and outlook," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 90(3), pages 1-13, March.
    8. Jewgeni H. Dshalalow, 1994. "First excess levels of vector processes," International Journal of Stochastic Analysis, Hindawi, vol. 7, pages 1-8, January.
    9. Lev Abolnikov & Jewgeni H. Dshalalow, 1992. "A first passage problem and its applications to the analysis of a class of stochastic models," International Journal of Stochastic Analysis, Hindawi, vol. 5, pages 1-15, January.
    10. Joseph Abate & Ward Whitt, 2006. "A Unified Framework for Numerically Inverting Laplace Transforms," INFORMS Journal on Computing, INFORMS, vol. 18(4), pages 408-421, November.
    11. Jewgeni H. Dshalalow & Jean-Baptiste Bacot, 2001. "On functionals of a marked Poisson process observed by a renewal process," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 26, pages 1-10, January.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ponta, Linda & Trinh, Mailan & Raberto, Marco & Scalas, Enrico & Cincotti, Silvano, 2019. "Modeling non-stationarities in high-frequency financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 521(C), pages 173-196.
    2. Aleksejus Kononovicius & Vygintas Gontis, 2019. "Approximation of the first passage time distribution for the birth-death processes," Papers 1902.00924, arXiv.org.
    3. Jaros{l}aw Klamut & Tomasz Gubiec, 2018. "Directed Continuous-Time Random Walk with memory," Papers 1807.01934, arXiv.org.
    4. Sokolov, Andrey & Melatos, Andrew & Kieu, Tien, 2010. "Laplace transform analysis of a multiplicative asset transfer model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(14), pages 2782-2792.
    5. A. Baykal Hafızoğlu & Esma S. Gel & Pınar Keskinocak, 2013. "Expected Tardiness Computations in Multiclass Priority M / M / c Queues," INFORMS Journal on Computing, INFORMS, vol. 25(2), pages 364-376, May.
    6. Dassios, Angelos & Li, Luting, 2020. "Explicit asymptotic on first passage times of diffusion processes," LSE Research Online Documents on Economics 103087, London School of Economics and Political Science, LSE Library.
    7. Dassios, Angelos & Qu, Yan & Zhao, Hongbiao, 2018. "Exact simulation for a class of tempered stable," LSE Research Online Documents on Economics 86981, London School of Economics and Political Science, LSE Library.
    8. David Landriault & Bin Li & Hongzhong Zhang, 2014. "On the Frequency of Drawdowns for Brownian Motion Processes," Papers 1403.1183, arXiv.org.
    9. Leippold, Markus & Vasiljević, Nikola, 2017. "Pricing and disentanglement of American puts in the hyper-exponential jump-diffusion model," Journal of Banking & Finance, Elsevier, vol. 77(C), pages 78-94.
    10. Illés Horváth & András Mészáros & Miklós Telek, 2020. "Numerical Inverse Transformation Methods for Z-Transform," Mathematics, MDPI, vol. 8(4), pages 1-18, April.
    11. Scalas, Enrico & Gallegati, Mauro & Guerci, Eric & Mas, David & Tedeschi, Alessandra, 2006. "Growth and allocation of resources in economics: The agent-based approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(1), pages 86-90.
    12. Runhuan Feng & Hans W. Volkmer, 2015. "Conditional Asian Options," Papers 1505.06946, arXiv.org.
    13. Mor Harchol-Balter, 2021. "Open problems in queueing theory inspired by datacenter computing," Queueing Systems: Theory and Applications, Springer, vol. 97(1), pages 3-37, February.
    14. D’Amico, Guglielmo & Janssen, Jacques & Manca, Raimondo, 2009. "European and American options: The semi-Markov case," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(15), pages 3181-3194.
    15. Pan, Aiqiang & McCartney, John S. & Lu, Lin & You, Tian, 2020. "A novel analytical multilayer cylindrical heat source model for vertical ground heat exchangers installed in layered ground," Energy, Elsevier, vol. 200(C).
    16. Bolster, Diogo & Benson, David A. & Singha, Kamini, 2017. "Upscaling chemical reactions in multicontinuum systems: When might time fractional equations work?," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 414-425.
    17. Gatto, R., 2018. "Saddlepoint approximation to the distribution of the total distance of the von Mises–Fisher continuous time random walk," Applied Mathematics and Computation, Elsevier, vol. 324(C), pages 285-294.
    18. Jiang, Zhi-Qiang & Chen, Wei & Zhou, Wei-Xing, 2009. "Detrended fluctuation analysis of intertrade durations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(4), pages 433-440.
    19. Markus Leippold & Nikola Vasiljević, 2020. "Option-Implied Intrahorizon Value at Risk," Management Science, INFORMS, vol. 66(1), pages 397-414, January.
    20. Villarroel, Javier & Montero, Miquel, 2009. "On properties of continuous-time random walks with non-Poissonian jump-times," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 128-137.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:10:p:1148-:d:557834. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.