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Stationary Markovian Processes with Long-Range correlations and Ergodicity breaking

Observatory of Complex Systems. http://lagash.dft.unipa.it. Stationary Markovian Processes with Long-Range correlations and Ergodicity breaking. Salvatore Miccichè with Fabrizio Lillo, Rosario N. Mantegna. Università degli Studi di Palermo, Dipartimento di Fisica e Tecnologie Relative.

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Stationary Markovian Processes with Long-Range correlations and Ergodicity breaking

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  1. Observatory of Complex Systems http://lagash.dft.unipa.it Stationary Markovian Processes with Long-Range correlations and Ergodicity breaking Salvatore Miccichè with Fabrizio Lillo, Rosario N. Mantegna Università degli Studi di Palermo, Dipartimento di Fisica e Tecnologie Relative INFORMAL WORKSHOP onFokker-Planck equations, algebraic correlations, long range correlations, and related questionsENS - Lyon, 29-30 March 2005

  2. Stationary Markovian Processes with Long-Range correlations and Ergodicity breaking Motivations Volatility in Financial Markets Time Series:Persistencies Volatility Clustering Empirical pdf: Lognormal - for intermediate values of volatility Power law - for large values of volatility (  4.8) Empirical Autocorrelation: Long-Range correlated process non-exponential asymptotic decay. Power-law (  0.3) ? Empirical Leverage: single exponentialfitting curve

  3. Stationary Markovian Processes with Long-Range correlations and Ergodicity breaking Motivations Stochastic Volatility Models discrete time •ARCH-GARCH, (Engle, Granger, …) continuous time • based on Langevin stochastic differential equations (with linear mean- reverting drift coefficient (Hull-White, Heston, Stein-Stein., …). • based on multifractality (Muzy et al., … ) • based on fractional Brownian motion (Sircar et al., … ) • …

  4. Stationary Markovian Processes with Long-Range correlations and Ergodicity breaking Aim of our research 1) Characterize the stationaryMarkovian stochastic processes wherethere is not one single characteristic time-scales. 2) Rather, we are interested in stationary Markovian stochastic processes withmany characteristic time-scales. 3) Moreover, we are interested in stationary Markovian stochastic processes with aninfinite ( how infinite? ) number of time-scales. 4)Explicit form of the autocorrelation function. Description in terms of Langevin stochastic differential equations. NO memory terms, NO fractional derivatives, …

  5. Stationary Markovian Processes with Long-Range correlations and Ergodicity breaking Outline PART I - stochastic processes Characterizestationary Markovian stochastic processes with multiple/infinite time-scales. Links with hamiltonian models PART II - ergodicity breaking Ergodicity breaking & autocorrelation function Ergodicity breaking & moments/FPTD PART III - stochastic volatility Use these results to build up a Markovian stochastic volatility model which incorporates the long-range memory of volatility.

  6. Stationary Markovian Stochastic Processes with Multiple Time-Scales Part IStationary Markovian Stochastic Processes with Multiple Time-Scales References [1] A. Schenzle, H. Brand, Phys. Rev. A, 20(4), 1628, (1979) [2] M. Suzuki, K. Kaneko, F. Sasagawa, Prog. Theor. Phys., 65(3), 828, (1981) [3] J. Farago, Europhys. Lett., 52(4), 379, (2000) [4] F. Lillo, S. Miccichè, R. N. Mantegna, cond-mat, 0203442, (2002)

  7. Stationary Markovian Stochastic Processes with Multiple Time-Scales Outline A) TOOL: We study the Autocorrelation function of a stochastic process described by a non-linear Langevin equation.METHODOLOGY: relationship between the Fokker-Planck equation and the Schrödinger equation with a potential VS. B) A simple example of the methodology used: the Ornstein-Uhlenbeck process. C) How the spectral properties of the (quantum) potential VS affect the structure of the autocorrelation function: processes with multiple time scales processes with infinite time scales PATH: from exponential to non exponential autocorrelation. END-POINT: power-law? i.e.Long-Range Correlatedprocesses ?

  8. Stationary Markovian Stochastic Processes with Multiple Time-Scales Nonlinear Langevin and Fokker-Planck Equations Nonlinear Langevin Equation Ito / Stratonovich prescription Fokker-Planck Equation AutoCorrelation function ()/AutoCovariance function R() Linear drift h=- x=> Exponential Autocorrelation exp(-  )

  9. Stationary Markovian Stochastic Processes with Multiple Time-Scales Markovian Property Chapman-Kolmogorov Equation: In the context ofcontinuous-timestochastic processes, this is the definition of a Markovian stochastic process we will consider.

  10. Stationary Markovian Stochastic Processes with Multiple Time-Scales Relationship Fokker-Planck / Schrödinger Hereafter we will consider the case of Additive Noise: g(x)=1 Stationary solution of Fokker-Planck eqn. Schrödinger equation quantum potential Stationarity is ensured if there exists a normalizable eigenfunction 0(x)corresponding to the eigenvalue E0=0.

  11. Stationary Markovian Stochastic Processes with Multiple Time-Scales Relationship Fokker-Planck / Schrödinger The validity of this methodology is based upon the assumption that i.e. the eigenfunctions {0, n, E} are a COMPLETEset of eigenfunctions in the SPACE of INTEGRABLE functions L1. Analogously, the eigenfunctions {0, n, E} must be a COMPLETEset of eigenfunctions in the SPACE of SQUARE-INTEGRABLE functions L2.

  12. Stationary Markovian Stochastic Processes with Multiple Time-Scales Relationship Fokker-Planck / Schrödinger Completeness in L2is equivalent to (1) which implies: (2) Does completeness in L2 imply completeness in L1 ?? Is (2) enough to ensurecompleteness in L1??

  13. Stationary Markovian Stochastic Processes with Multiple Time-Scales Relationship Fokker-Planck / Schrödinger 2-point probability density AutoCovariance Function odd eigenfunctions

  14. Stationary Markovian Stochastic Processes with Multiple Time-Scales AutoCorrelation Function Discrete Spectrum only Therefore, in order to have not-exponential AutoCovariance function we need to introduce a continuum part in the spectrum. discrete how infinite? continuum

  15. Stationary Markovian Stochastic Processes with Multiple Time-Scales Ornstein-Uhlenbeck process In this case we have one single time-scale. In this case we have one single time-scale. Linear driftExponential autocorrelation expected With this choice of variables, the Schrödinger equation is the same as the one associated to the harmonic oscillator potential, which is completely solvable: ground state discrete eigenfunctionsHn Hermite polynomials

  16. Stationary Markovian Stochastic Processes with Multiple Time-Scales Ornstein-Uhlenbeck process We can compute the AutoCorrelation Function due to properties of Hermite polynomials AutoCovariance Variance AutoCorrelation Time-Scale T=1/

  17. Stationary Markovian Stochastic Processes with Multiple Time-Scales The Square Well In this case we have a numerable set of time-scales. This is the Infinite Square Well problem. It is completely solvable: ground state antisymmetricdiscrete eigenfunctionsn odd Tn=1/En

  18. Stationary Markovian Stochastic Processes with Multiple Time-Scales The delta-like potential In this case we have a continuum spectrum: GAP This is the -like Wellproblem. It is completely solvable: ground state antisymmetriccontinuum eigenfunctionsn odd GAP

  19. Stationary Markovian Stochastic Processes with Multiple Time-Scales The delta-like potential All functionscE can be obtained analytically.Also the further integration to get R()can be performed analytically. “Completeness” The picture shows a comparison between the theoretical results and a numerical simulation of the stochastic process, i.e. a numerical integration of the Langevin equation. The numerical simulation is performed starting from the only knowledge of the drift coefficient h(x). Therefore, it is completely independent from the theoretical procedure used to obtain the autocovariance function.

  20. Stationary Markovian Stochastic Processes with Multiple Time-Scales A quantum potential with gap In this case we have a continuum spectrum: GAP This potential is still completely solvable.Eigenfunctions are expressed in terms of Bessel functions J(.)and Y(.). GAP

  21. Stationary Markovian Stochastic Processes with Multiple Time-Scales A quantum potential with gap cEanalytically.R() numerically. “Completeness“ R() exp{-V22} The cut-off effect given by the the exponential term gets smaller along with V2  0.

  22. Stationary Markovian Stochastic Processes with Multiple Time-Scales A quantum potential without gap So far: continuum spectrumnon exponential autocorrelation T is related to the inverse of the energy gap. Single/Multiple(OU)Time-ScalesInfinite(shift)Time-Scales. What about reducing theenergy gap to zero ?  V1 -V0 NO GAP

  23. Stationary Markovian Stochastic Processes with Multiple Time-Scales Long-Range Correlated processes: CHIMERA Eigenfunctions x>L Normalizationconditions This condition is fulfilled if: It is worth noting that this is the only way to fulfill the following condition:

  24. Stationary Markovian Stochastic Processes with Multiple Time-Scales Long-Range Correlated processes: CHIMERA We can prove that, for INVERSE SQUARED POTENTIALS : All functionscE can be obtained analytically. Further integration to get R()can only be performed numerically. One can only show that asymptotically: Anomalous Diffusion 3<<5 <1

  25. Stationary Markovian Stochastic Processes with Multiple Time-Scales Long-Range Correlated processes: CHIMERA

  26. Stationary Markovian Stochastic Processes with Multiple Time-Scales Long-Range Correlated processes: CHIMERA ?? continuum spectrum + zero gap +VS  x -2long-range processesi.e.not integrableautocorrelation function Integrable Short-range correlated  >1 Short-range correlated Not Integrable Long-range correlatedAnomalous diffusion  <1

  27. Stationary Markovian Stochastic Processes with Multiple Time-Scales Long-Range Correlated processes: CHIMERA f(x) x Is everything OK ? Almost! 1)Yet we do not have a proof of completeness !! 2)Simulationsof the Autocorrelation are “DIFFICULT”

  28. Stationary Markovian Stochastic Processes with Multiple Time-Scales Long-Range Correlated processes: CHIMERA Is there agreement between simulations and numerical integrations of the eigenfunctions??

  29. Stationary Markovian Stochastic Processes with Multiple Time-Scales Long-Range Correlated processes: CHIMERA Is there agreement between simulations and numerical integrations of the eigenfunctions??

  30. Stationary Markovian Stochastic Processes with Multiple Time-Scales Another quantum potential without gap - I Consider the process: Do we get power-law decaying correlations? NO

  31. Stationary Markovian Stochastic Processes with Multiple Time-Scales Another quantum potential without gap - II Consider the process: Red curves are CHIMERA. Black curves are a=0.5

  32. Stationary Markovian Stochastic Processes with Multiple Time-Scales Another quantum potential without gap - II Limit a=0 ?

  33. Stationary Markovian Stochastic Processes with Multiple Time-Scales Hamiltonian models: mean field theory Consider the Hamiltonian [4] that describes a set of N particles governed by long-range interactions. We (TD, FB, SR, …) are interested in deriving a Fokker-Planck equation which describes the stochastic process of 1 particle in interaction with a bath of N-1 particles in equilibrium [4]. References [4] F. Bouchet, PRE, 70, 036113, (2004) [5] F. Bouchet, T. Dauxois cond-mat, 0407703,(2004) - submitted to PRL

  34. Stationary Markovian Stochastic Processes with Multiple Time-Scales Hamiltonian models: mean field theory In [4] Eq. 10 Eq. 9 In [5] Eq. 11 Eq. 12 where f0(p) is some “given” equilibrium distribution of the N particles. This FP leads to “… unsual algebraic correlation laws and to anomalous diffusion …”. QUESTIONS are:i) what are the (“physics”) differences between the two FPs?ii) why you consider =3, i.e. processes for which variance is not well defined?iii) how do you write the cross-correlations between particles ?

  35. Stationary Markovian Stochastic Processes with Multiple Time-Scales Conclusions Quantum Potential VS AutoCorrelation • Discrete SpectrumNumerable set of Time-Scales.Integrable AutoCorrelation.. Short-Range Correlated Processes. • Continuum Spectrum with GapInfinite set of Time-Scales. Integrable AutoCorrelation.. Short-Range Correlated Processes. • Continuum Spectrum without GapInfinite set of Time-Scales. Not-Integrable AutoCorrelation.. VS = V1/xlog(x)a=2, a=0Long-Range Correlated Processes.      “Peculiarity!! ” “Completeness ??”

  36. Ergodicity breaking Part IIErgodicity Breaking References [1] E. Lutz, PRL, 93, 190602, (2004) [2] S. Miccichè, F. Lillo, R. N. Mantegna, in preparation [3] J.-P. Bouchaud, J. Phys. I France, 2, 1705, (1992) [4] G. Bel, E. Barkai, cond-mat/0502154

  37. Ergodicity breaking Definition of Ergodicity A process is said to be Ergodic in the “Mean Square sense” iff (?) We will always consider: An(t)=x(t)n

  38. Ergodicity breaking Ergodicity for CHIMERA Ergodicity for CHIMERA holds for any > 2n+1 n=1 >3n=2 >5n=3 >7n=4 >9  =4 n=1 =5 n=1 =6 n=1,2 =7 n=1,2 Moments for CHIMERA are well defined for any > n+1 n=1 >2n=2 >3n=3 >4n=4 >5  =4 n=1,2 =5 n=1,2,3 =6 n=1,2,3,4 =7 n=1,2,3,4,5

  39. Ergodicity breaking Ergodicity for CHIMERA Moments of FPTD g2(t) for CHIMERA are well defined for any  > 2m-1 m=1 >1m=2 >3m=3 >5m=4 >7  =4 m=1,2 =5 m=1,2 =6 m=1,2,3 =7 m=1,2,3 Value of n ERGO MOMENTS FPTDn=1 yes yes yesn=2 >5 yes yesn=4 >9>5>7...

  40. Ergodicity breaking Ergodicity & Autocorrelation function Let us consider n=2:   Ergo Moments FPTD =4 =0.5 long-rangeno up to 3rd up to 2nd =4.8 =0.9 long-rangeno up to 3rd up to 2nd =5.1 =1.1 short-rangeyes up to 4thup to 3rd =6 =1.5 short-rangeyes up to 4th up to 3rd TIME-AVERAGE ENSEMBLE-AVERAGE xi(0) are distributed according to the stationary pdf.

  41. Ergodicity breaking Ergodicity & Autocorrelation function

  42. Ergodicity breaking Ergodicity & Autocorrelation function

  43. Ergodicity breaking Multiplicative CHIMERA drift pdf Auto-correlation Coordinatetransformation(asymptotically)

  44. Ergodicity breaking Multiplicative CHIMERA

  45. Ergodicity breaking Multiplicative CHIMERA Ergodicity for Multiplicative CHIMERA holds for any  > 2n+1  =4 n=1 =5 n=1 =6 n=1,2 =7 n=1,2 Short-Range & non-ergodic Long-Range & Ergodic n=1 >3n=2  >5n=3  >7n=4  >9

  46. Ergodicity breaking Multiplicative CHIMERA =0.5 (long-range)  =5.5 (ergodic)  =-0.5

  47. Ergodicity breaking Multiplicative CHIMERA =1.5 (short-range)  =3.5 (NON-ergodic)  =+0.5

  48. Ergodicity breaking Tentative Conclusions 1) Ergodicity  Short-Range Correlation Non-Ergodicity  Long-Range Correlation Is true only true for Stationary Markovian processes with additive noise For Stationary Markovian processes with multiplicative noise one might have: 2) Non-Ergodicity & Short-Range Correlation Ergodicity  Long-Range Correlation 3)QUESTIONis: what is the intimate source of Ergodicity Breaking ? MOMENTS of pdf are diverging ? or MOMENTS of FPTD are diverging - Sojourn times [3]? In CHIMERA-like processes these features are both present. What about other (non markovian ? ) processes [4]?

  49. A Two-Region Stochastic Volatility Model Part IIIEconoPhysicsfrom -PHYSICS to ECONO- (?)

  50. A Two-Region Stochastic Volatility Model Issue 1)mean revertingOU-like driving processes Issue 2) long-range fBm Multifractal ... Macroscopic / Phenomenological Model

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