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### Stochastic Processes

Part I
• L1 (Time and Ensemble averaging)
• L2 (Joint and conditional probability distribution functions)
• L3  (Generating functions, kurtosis, skewness, correlation functions, stationary processes)
• L4  (Spectral Density)
• L5  ( Spectral Density - Fast Fourier Transformation (FFT ))
• L6-1 (Spectral Density - Maximum Entropy Method-1 )
• L6-2 (Spectral Density - Maximum Entropy Method-2, Yule-Walker Equations)
• L7 (Cumulants, Multidimensional Gaussian distribution, Wick theorem)
• L8 (Classification of stochastic processes, Markov processes, Chapman-Kolmogorov equation, Markov time or length scale, Langevin equation, Brownian motion)
• L9 (Synthesis of correlated Gaussian signal, Box-Muller method, Generation of long-range, scaling correlated signal, Fourier-Filtering Method)
• L10 (Random-Walk, Detrended Fluctuation Analysis(DFA), Detrended Moving Average (DMA))
• L11 (Rescaled-Range-Analysis (R/S), Multifractals and Singularity Spectrum)
• L12 (Multi-fractal Detrended Fluctuation Analysis(MFDFA) and Renyi exponents (or generalized fractal dimension D(q) ),Shuffled and Surrogate time series, Fourier Detrended Fluctuation Analysis(FDFA), Wavelet transformation-1)
• L13 (Windowed Fourier Transformation (WFT), Wavelet Transformation-2, Lipchitz Regularity, Wavelet Transform Modulus Maxima Method (WTMM), Comparison of Multi-fractal exponents of MFDFA and WTMM)
• L14 (Density Function Estimation: {Naive, Kernel, Sample distribution function, Nearest-Neighbors and Variable kernel estimators}, Chi-Square Test, Estimation of the Markov time or length scales via Chi-Square test, Kolmogorov-Simrnov Test, Kullback-Leibler entropy (distance), Anderson-Darling Test)
• L15-1 ,L15-2 ( Maximum Likelihood Function, Confidence Levels and Intervals)

part II

• L16 (Kramers-Moyal Forward and Backward Expansions, Formal Solution, Karamers-Moyal coefficents of Brownian Motion)
• L17 (Pawula Theorem, Stationary Solution of the Fokker-Planck Equation, Transition Probability Density for Small Times, Path-Integral Solutions of Time Dependent Fokker-Planck Equation)
• L18 (Wiener Process, Non-Differentiability of Sample Path, Independence of Increments, Ornestien-Uhlenbeck Process, The Method of Characteristics)
• L19 (First-Passage-Time and Its Moments in terms of Drift and Diffusion Coefficients, Escape Over a Potential Barrier, Level Crossing )
• L20 ( The Ito and Stratonovich Integrals, Nonparticipating Functions, Ito Relation, Fokker-Planck Equation of the General Langevin Equation, Kubo Oscillator, Black-Scholes Stochastic Differential Equation)

PART III

• L21 ( Scaling Functions and Scaling Invariance, Discrete Scale Invariance, Scaling Invariance in Stochastic Prosseses, Fractional Brownian Motion (FBM),Fractional Gaussian noise (FGn), The realation between the DFA exponent and Hurst exponent of FBM and FGn signals, Generation of FBM signal as an optimization problem)

The Pdf files of the lecture notes (in Persian) are avaliable under request.