Statistical Arbitrage in Algorithmic Trading: Stationarity, Cointegration, and Mean-Reverting Stochastic Processes With Python (Computational Mathematics Library)

It is written for practitioners who require precise assumptions, reproducible algorithms, and measurable error. The exposition is measure-theoretic and asymptotic so that stability, efficiency, and risk are established on a rigorous basis. Rather than including formal proofs, theorems are demonstrated with Python code. Each chapter concludes with end of chapter Python demonstrations and tests that connect formulas to execution. Coverage is complete for a statistical arbitrage pipeline built on stationarity and mean reversion. It is shown how to construct cointegrated spreads by rank conditions, projections, and reduced-rank regression. It is derived how to stabilize vector error-correction models through eigenstructure of the companion matrix. It is implemented how to calibrate Ornstein Uhlenbeck diffusions by exact discretization, MLE, and local to unity analysis. Continuous time cointegration is treated via quadratic covariation with discrete observation noise. State space models and the Kalman filter are obtained from first principles and used for online beta and latent spread estimation. To handle nonstationarity and structural change, regime switching diffusions with hidden Markov chains are estimated by EM with attention to identifiability and persistence. Fractional integration and long memory are treated by local Whittle and GPH with careful bandwidth control. Spectral density estimation, multitaper methods, and cross spectral analysis provide low frequency risk diagnostics. High dimensional selection of pairs and portfolios is performed by sparse canonical correlation with dependence aware validation. Covariance and precision matrices are stabilized by shrinkage and graphical methods with robust alternatives for heavy tails. Empirical process theory supports unit root and stationarity testing with self normalization and dependent bootstrap. Heavy tails are modeled by alpha stable innovations and Lévy driven OU processes, with robust M estimation protecting signals and leverage. First passage and hitting time calculations give expected trade duration and stop out probabilities. Optimal stopping and impulse control with transaction costs yield cost aware entry exit bands with smooth fit and numerical verification. Kelly growth is derived for stationary spreads with constraints that control drawdowns. Change point detection by generalized likelihood ratios, CUSUM, and Shiryaev Roberts is calibrated for dependent noise. Multiple testing across many spreads is controlled by Benjamini Hochberg, Benjamini Yekutieli, and adaptive q value procedures, with online control by LORD and SAFFRON under dependence. Online learning with drifting parameters is handled by stochastic approximation and filtering with regret bounds. Concentration inequalities under alpha mixing and matrix Freedman bounds provide nonasymptotic risk control. Semiparametric efficiency and Godambe information guide estimator choice, while quadratic form risk decomposition attributes variance to frequency bands, factors, and hedge errors.