Abstract
In this paper, we are addressing the old problem of long-term nonlinear autocorrelation function versus short-term linear autocorrelation function. As continuous-time random walk (CTRW) can describe almost all possible kinds of diffusion, it seems to be an excellent tool to use. To be more precise, for instance, CTRW can successfully describe the short-term negative autocorrelation of returns in high-frequency financial data (caused by the bid-ask bounce phenomena). We observe long-term autocorrelation of absolute values of returns. Can it also be described by the CTRW model? And maybe more importantly, to what extent can it be explained by the same phenomena? To refer to these questions, we propose a new directed CTRW model with memory. The canonical CTRW trajectory consists of spatial jumps preceded by waiting times. In directed CTRW, we consider the case with positive spatial jumps only. We take into account the memory in the model as each spatial jump depends on the previous one. This model, based on simple assumptions, allowed us to obtain the general formula covering most popular types of nonlinear autocorrelation functions.
Graphical abstract
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E.W. Montroll, G.H. Weiss, J. Math. Phys. 6, 167 (1965)
R. Kutner, J. Masoliver, Eur. Phys. J. B 90, 50 (2017)
H. Scher, E.W. Montroll, Phys. Rev. B 12, 2455 (1975)
G. Pfister, H. Scher, Adv. Phys. 27, 747 (1978)
E.W. Montroll, M.F. Schlesinger, in Nonequilibrium Phenomena II: From Stochastics to Hydrodynamics, edited by J. Lebowitz, E. Montroll (North-, Amsterdam, 1984), pp. 1–121
G. Weiss, in Fractals in Science (Springer, Berlin, 1994), pp. 119–162
J.P. Bouchaud, A. Georges, Phys. Rep. 195, 127 (1990)
D. Ben-Avraham, S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press, Cambridge, 2000)
R. Hilfer, Physica A 329, 35 (2003)
E. Barkai, Y.C. Cheng, J. Chem. Phys. 118, 6167 (2003)
C. Monthus, J.P. Bouchaud, J. Phys. A, Math. Gen. 29, 3847 (1996)
W. Dieterich, P. Maass, Solid State Ion. 180, 446 (2009)
N. Iyengar, C. Peng, R. Morin, A. Goldberger, L. Lipsitz, Am. J. Physiol. Regul. Integr. Comp. Physiol. 271, R1078 (1996)
J. Nelson, Phys. Rev. B 59, 15374 (1999)
M. Lomholt, K. Tal, R. Metzler, J. Klafter, Proc. Natl. Acad. Sci. 105, 11055 (2008)
G. Margolin, B. Berkowitz, J. Phys. Chem. B 104, 3942 (2000)
A. Helmstetter, D. Sornette, Phys. Rev. E 66, 061104 (2002)
H. Scher, G. Margolin, R. Metzler, J. Klafter, B. Berkowitz, Geophys. Res. Lett. 29, 5 (2002)
R. Hempelmann, in Anomalous Diffusion From Basics to Applications, edited by R. Kutner, A. Pȩkalski, K. Sznajd-Weron (Springer, Berlin, Heidelberg, 1999), pp. 247–252
L. Hufnagel, D. Brockmann, T. Geisel, Nature 439, 462 (2006)
E. Scalas, R. Gorenflo, F. Mainardi, Physica A 284, 376 (2000)
F. Mainardi, M. Raberto, R. Gorenflo, E. Scalas, Physica A 287, 468 (2000)
M. Raberto, E. Scalas, F. Mainardi, Physica A 314, 749 (2002)
E. Scalas, R. Gorenflo, F. Mainardi, Phys. Rev. E 69, 011107 (2004)
E. Scalas, Physica A 362, 225 (2006)
R. Kutner, F. Świtała, Quant. Financ. 3, 201 (2003)
J. Masoliver, M. Montero, G.H. Weiss, Phys. Rev. E 67, 021112 (2003)
P. Repetowicz, P. Richmond, Physica A 344, 108 (2004). (Applications of Physics in Financial Analysis 4 (APFA4))
J. Masoliver, M. Montero, J. Perelló, G.H. Weiss, J. Econ. Behav. Organ. 61, 577 (2006)
J. Masoliver, M. Montero, J. Perello, G.H. Weiss, Physica A 379, 151 (2007)
R. Kutner, Phys. A, Stat. Mech. Appl. 314, 786 (2002)
E. Scalas, in The Complex Networks of Economic Interactions (Springer, Berlin, Heidelberg, 2006), pp. 3–16
J. Perelló, J. Masoliver, A. Kasprzak, R. Kutner, Phys. Rev. E 78, 036108 (2008)
T. Gubiec, R. Kutner, Eur. Phys. J. B 90, 228 (2017)
T. Gubiec, R. Kutner, Phys. Rev. E 82, 046119 (2010)
A. Kasprzak, R. Kutner, J. Perelló, J. Masoliver, Eur. Phys. J. B 76, 513 (2010)
R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)
H. Scher, M.F. Shlesinger, J.T. Bendler, Phys. Today 44, 26 (1991)
J.W. Haus, K.W. Kehr, Phys. Rep. 150, 263 (1987)
K. Kehr, R. Kutner, K. Binder, Phys. Rev. B 23, 4931 (1981)
R. Kutner, J. Phys. C: Solid State Phys. 18, 6323 (1985)
T. Gubiec, R. Kutner, Acta Phys. Pol. A 117, 669 (2010)
M. Montero, J. Masoliver, Phys. Rev. E 76, 061115 (2007)
M. Montero Torralbo, Phys. Rev. E 84, 051139 (2011)
V. Tejedor, R. Metzler, J. Phys. A: Math. Theor. 43, 082002 (2010)
M. Magdziarz, R. Metzler, W. Szczotka, P. Zebrowski, Phys. Rev. E 85, 051103 (2012)
A.V. Chechkin, M. Hofmann, I.M. Sokolov, Phys. Rev. E 80, 031112 (2009)
J.H. Jeon, N. Leijnse, L.B. Oddershede, R. Metzler, New J. Phys. 15, 045011 (2013)
R. Tsay, Analysis of Financial Time Series, 2nd edn., Wiley Series in Probability and Statistics (Wiley-, Hoboken, NJ, 2005)
R. Cont, Quant. Financ. 1, 223 (2001)
R. Cont, in Fractals in Engineering, edited by J. Lévy-Véhel, E. Lutton (Springer, London, 2005), pp. 159–179
J.K.E. Tunaley, Phys. Rev. Lett. 33, 1037 (1974)
J.K.E. Tunaley, J. Stat. Phys. 11, 397 (1974)
T. Gubiec, M. Wiliński, Physica A 432, 216 (2015)
J. Hasbrouck, Empirical Market Microstructure: The Institutions, Economics, and Econometrics of Securities Trading (Oxford University Press, Oxford, 2007)
M.M. Dacorogna, R. Gencay, U. Muller, R.B. Olsen, O.V. Pictet, An Introduction to High Frequency Finance (Academic Press, New York, 2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://doi.org/creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Klamut, J., Gubiec, T. Directed continuous-time random walk with memory. Eur. Phys. J. B 92, 69 (2019). https://doi.org/10.1140/epjb/e2019-90453-y
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2019-90453-y