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Data Driven Study of Stochastic Processes with Non-Vanishing Higher-Order Kramers-Moyal Coefficients

Ⅿohebbi, Hassan | 2024

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 57242 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Rahimi Tabar, Mohammad Reza
  7. Abstract:
  8. According to the Pawula theorem, we know that stochastic processes with non-vanishing higher-order Kramers-Moyal coefficients cannot be described by the Fokker-Planck and Langevin equation. Using rotating Gaussian white noise, we define generalized Brownian motion of different orders, also known as fractional Brownian motion. Similar to the Langevin equation, which is used for the random behavior of a stochastic process using standard Brownian motion, we also use a set of generalized Brownian motions for the random behavior of a stochastic process that has non-vanishing higher-order Kramers-Moyal coefficients of higher orders. Given that these noises are complex-valued, we define the stochastic process in the complex plane. For an empirical time series that has non-vanishing higher-order Kramers-Moyal coefficients, we can calculate the Kramers-Moyal coefficients of the stochastic differential equation such that the projection of the complex stochastic process on the real axis is a process that has Kramers-Moyal coefficients equal to Kramers-Moyal coefficients of the empirical time series. Therefore, for an empirical time series, we can write a stochastic differential equation using Kramers-Moyal coefficients of any desired order
  9. Keywords:
  10. Stochastic Process ; Brownian motion ; Time Series ; Kramers-Moyal Coefficient ; Langevin Equation ; Fokker-Planck Equation ; Non-Vanishing Kramers-Moyal Coefficients ; Order N Gaussian White Noise Of ; Generalized Ito Lemma ; Generalized Fokker-Planck Equation

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