Spurious Memory in Non-Equilibrium Stochastic Models of Imitative Behavior

The origin of the long-range memory in the non-equilibrium systems is still an open problem as the phenomenon can be reproduced using models based on Markov processes. In these cases a notion of spurious memory is introduced. A good example of Markov processes with spurious memory is stochastic process driven by a non-linear stochastic differential equation (SDE). This example is at odds with models built using fractional Brownian motion (fBm). We analyze differences between these two cases seeking to establish possible empirical tests of the origin of the observed long-range memory. We investigate probability density functions (PDFs) of burst and inter-burst duration in numerically obtained time series and compare with the results of fBm. Our analysis confirms that the characteristic feature of the processes described by a one-dimensional SDE is the power-law exponent $3/2$ of the burst or inter-burst duration PDF. This property of stochastic processes might be used to detect spurious memory in various non-equilibrium systems, where observed macroscopic behavior can be derived from the imitative interactions of agents.

[1]  Xavier Gabaix,et al.  Price fluctuations, market activity and trading volume , 2001 .

[2]  Kevin E. Bassler,et al.  Hurst exponents, Markov processes, and fractional Brownian motion , 2006, cond-mat/0609671.

[3]  T. Lux,et al.  Estimation of Agent-Based Models: The Case of an Asymmetric Herding Model , 2005 .

[4]  A. Lo.  Long-Term Memory in Stock Market Prices , 1989 .

[5]  C. Tsallis.  Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World , 2009 .

[6]  S. Fortunato,et al.  Statistical physics of social dynamics , 2007, 0710.3256.

[7]  Vygintas Gontis,et al.  Consentaneous Agent-Based and Stochastic Model of the Financial Markets , 2014, PloS one.

[8]  M. Yor,et al.  Mathematical Methods for Financial Markets , 2009 .

[9]  Julius Ruseckas,et al.  Continuous transition from the extensive to the non-extensive statistics in an agent-based herding model , 2014, 1404.0856.

[10]  Ding,et al.  Distribution of the first return time in fractional Brownian motion and its application to the study of on-off intermittency. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Raúl Toral,et al.  Markets, Herding and Response to External Information , 2015, PloS one.

[12]  Vygintas Gontis,et al.  Long-range memory model of trading activity and volatility , 2006 .

[13]  V. Gontis,et al.  Modeling long-range memory trading activity by stochastic differential equations , 2006, physics/0608036.

[14]  V. Gontis,et al.  Burst and inter-burst duration statistics as empirical test of long-range memory in the financial markets , 2017, 1701.01255.

[15]  Thomas Mikosch,et al.  Long range dependence effects and ARCH modeling , 2003 .

[16]  L. Onsager.  Crystal statistics. I. A two-dimensional model with an order-disorder transition , 1944 .

[17]  B Kaulakys,et al.  1/f Noise from nonlinear stochastic differential equations. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  V. Plerou,et al.  A theory of power-law distributions in financial market fluctuations , 2003, Nature.

[19]  Michel Mandjes,et al.  ON SPECTRAL SIMULATION OF FRACTIONAL BROWNIAN MOTION , 2003, Probability in the Engineering and Informational Sciences.

[20]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[21]  B Kaulakys,et al.  Point process model of 1/f noise versus a sum of Lorentzians , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  J. Pitman.  Combinatorial Stochastic Processes , 2006 .

[23]  Alan Kirman,et al.  Ants, Rationality, and Recruitment , 1993 .

[24]  V. Gontis,et al.  A long-range memory stochastic model of the return in financial markets , 2009, 0901.0903.

[25]  鈴木 増雄.  Time-Dependent Statistics of the Ising Model , 1965 .

[26]  B. Kaulakys,et al.  Scaling properties of signals as origin of 1/f noise , 2014, 1402.2523.

[27]  Simone Alfarano,et al.  Network structure and N-dependence in agent-based herding models , 2009 .

[28]  S. Redner.  A guide to first-passage processes , 2001 .

[29]  Vygintas Gontis,et al.  Modelling financial markets by the multiplicative sequence of trades , 2004, ArXiv.

[30]  H. Risken.  Fokker-Planck Equation , 1984 .

[31]  M Ausloos,et al.  Dynamical model and nonextensive statistical mechanics of a market index on large time windows. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  Maxi San Miguel,et al.  Is the Voter Model a model for voters? , 2013, Physical review letters.

[33]  V. Gontis,et al.  THE CLASS OF NONLINEAR STOCHASTIC MODELS AS A BACKGROUND FOR THE BURSTY BEHAVIOR IN FINANCIAL MARKETS , 2012, 1201.3083.

[34]  R. Holley,et al.  Ergodic Theorems for Weakly Interacting Infinite Systems and the Voter Model , 1975 .

[35]  西森 秀稔.  Statistical physics of spin glasses and information processing : an introduction , 2001 .

[36]  B. Kaulakys,et al.  1/f noise from the nonlinear transformations of the variables , 2015, 1512.04298.

[37]  Constantino Tsallis,et al.  Introduction to Nonextensive Statistical Mechanics and Thermodynamics , 2003 .

[38]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[39]  P. Clifford,et al.  A model for spatial conflict , 1973 .

[40]  Walter Willinger,et al.  Stock market prices and long-range dependence , 1999, Finance Stochastics.

[41]  J. McCauley,et al.  Markov processes, Hurst exponents, and nonlinear diffusion equations: With application to finance , 2006, cond-mat/0602316.

[42]  青木 正直,et al.  Reconstructing macroeconomics : a perspective from statistical physics and combinatorial stochastic processes , 2007 .

[43]  Boris Podobnik,et al.  Stochastic model of financial markets reproducing scaling and memory in volatility return intervals , 2015, 1507.05203.

[44]  Aleksejus Kononovicius,et al.  Empirical Analysis and Agent-Based Modeling of the Lithuanian Parliamentary Elections , 2017, Complex..

[45]  A. J. McKane,et al.  Stochastic models of evolution in genetics, ecology and linguistics , 2007, cond-mat/0703478.

[46]  Andrew J. Patton,et al.  What good is a volatility model? , 2001 .

[47]  Vygintas Gontis,et al.  Agent based reasoning for the non-linear stochastic models of long-range memory , 2011, 1106.2685.

[48]  T. Liggett,et al.  Stochastic Interacting Systems: Contact, Voter and Exclusion Processes , 1999 .

[49]  B Kaulakys,et al.  Tsallis distributions and 1/f noise from nonlinear stochastic differential equations. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[50]  C. Lanouar.  A VARIETIES OF SPURIOUS LONG MEMORY PROCESS , 2006 .

[51]  M. Ausloos,et al.  Applications of statistical physics to economic and financial topics , 1999 .

[52]  Vygintas Gontis,et al.  Three-state herding model of the financial markets , 2012, 1210.1838.