Langevin equations from time series.

We discuss the link between the approach to obtain the drift and diffusion of one-dimensional Langevin equations from time series, and Pope and Ching's relationship for stationary signals. The two approaches are based on different interpretations of conditional averages of the time derivatives of the time series at given levels. The analysis provides a useful indication for the correct application of Pope and Ching's relationship to obtain stochastic differential equations from time series and shows its validity, in a generalized sense, for nondifferentiable processes originating from Langevin equations.

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

[2]  S. Pope,et al.  Stationary probability density functions: An exact result , 1993 .

[3]  Ching.  Probability densities of turbulent temperature fluctuations. , 1993, Physical review letters.

[4]  Valeri Kontorovich,et al.  Stochastic differential equations: an approach to the generation of continuous non-Gaussian processes , 1995, IEEE Trans. Signal Process..

[5]  J. Peinke,et al.  Description of a Turbulent Cascade by a Fokker-Planck Equation , 1997 .

[6]  J. Peinke,et al.  Statistical properties of a turbulent cascade , 1997 .

[7]  Serguei Primak,et al.  On the generation of the baseband and narrowband non-Gaussian processes , 1998, IEEE Trans. Signal Process..

[8]  Characterization of stationary distributions using conditional expectations , 1999 .

[9]  I. Sokolov.  Relation between the probability density and other properties of a stationary random process. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  Jan Raethjen,et al.  Extracting model equations from experimental data , 2000 .

[11]  The first 20 min in the Hong Kong stock market , 2000, cond-mat/0006145.

[12]  Gradisek,et al.  Analysis of time series from stochastic processes , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  Primak.  Generation of compound non-gaussian random processes with a given correlation function , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[14]  Modelling High-frequency Economic Time Series , 2000, cond-mat/0007267.

[15]  Friedrich,et al.  How to quantify deterministic and random influences on the statistics of the foreign exchange market , 1999, Physical review letters.

[16]  H Kantz,et al.  Indispensable finite time corrections for Fokker-Planck equations from time series data. , 2001, Physical review letters.

[17]  H. Kantz,et al.  Nonequilibrium physics meets time series analysis: Measuring probability currents from data , 2003 .

[18]  Langevin modelling of high-frequency Hang-Seng index data , 2003 .