Stochastic Process

This page contains resources about Stochastic Processes, Stochastic Systems, Random Processes and Random Fields.

More specific information is included in each subfield.

Subfields and Concepts

See Category:Stochastic Processes for some of its subfields.

• Discrete-time Stochastic Processes
• Continuous-time Stochastic Processes
• State Space
  • Discrete
  • Continuous
• Weak-sense Stationary Process / Wide-sense Stationary Process
• Strictly Stationary Process / Strongly Stationary Process
• Ergodicity / Ergodic Process
  • Mean-ergodic
  • Autocovariance-ergodic
  • Ergodic in the wide sense
• Time domain
  • Autocorrelation Function
• Frequency domain
  • Power Spectral Density / Power spectrum / Spectrum
  • Cramér Spectral Representation
• Wold Decomposition Theorem / Wold Representation Theorem
• Time Series Processes (Discrete-time and Continuous State Space)
  • Autoregressive (AR) Process
  • Moving Average (MA) Process
  • ARMA Process
• Latent Variable Models (i.e. Partially Observed Probabilistic Models)
  • Continuous Latent Variable Models
  • Discrete Latent Variable Models
• Stochastic Dynamical System
• Random Dynamical System
• Random Graphs
• Random Fields
  • Markov Random Field / Markov Network
    • Gibbs Random Field
    • Gaussian MRF / Multivariate Gaussian distribution (not to be confused with Gaussian Random Field)
  • Gaussian Random Field
• Markov Models
  • Discrete-time Markov Chain (Discrete-time and Discrete State Space)
  • Discrete-time Harris Chain (Discrete-time and Continuous State Space)
  • Continuous-time Markov Chain / Continuous-time Markov Process / Markov Jump Process
  • Continuous-time Stochastic Process with the Markov property (e.g. Wiener Process)
  • Hidden Markov Model
  • Markov Decision Process
  • Partially Observable Markov Decision Process
  • Hierarchical Markov Models
• Gaussian Process
• Gauss–Markov Process / AR Process
• Ornstein-Uhlenbeck Process / Stationary Gauss–Markov Process
• Wiener Process / Brownian Motion (Continuous-time and Continuous State Space)
• Geometric Brownian Motion
• Harmonic Process (e.g. Sinusoidal Model)
• Innovations Process
• Queues
• Martingales
• Jump Process
• Point Process
  • Cox Point Process
  • Poisson Process
• Dirichlet Process
• Pitman–Yor Process
• Chinese Restaurant Process
• Indian Buffet Process
• Lévy Process
• Bernoulli Process
• Pólya's Urn Process
• Hoppe's Urn Process
• Stick Breaking Process
• Girsanov Transformation / Girsanov Theorem (in Probability Theory)
• Stochastic Calculus (used in Computational Finance)
  • Itô Calculus
  • Itô's Lemma
  • Semimartingale

Online Courses

Video Lectures

• Discrete Stochastic Processes by Robert Gallager

Lecture Notes

• Stochastic Processes by Cosma Shalizi
• Introduction to Stochastic Systems by Maxim Raginsky
• Introduction to Stochastic Processes by Hao Wu
• Advanced Stochastic Processes by David Gamarnik
• Stochastic Systems by Florian Herzog
• Stochastic Calculus by Jonathan Goodman
• Stochastic Calculus by Michael R. Tehranchi
• Stochastic Calculus by Fabrice Baudoin
• Stochastic Calculus by Alan Bain

Books and Book Chapters

See Amazon and Google-Books for more books.

• Hajek, B. (2015). Random Processes for Engineers. Cambridge University Press.
• Pavliotis, G. A. (2014). Stochastic Processes and Applications. Springer.
• Stark, H., Woods, J. W., Thilaka, B., & Kumar, A. (2012). Probability, statistics, and random processes for engineers. 4th Ed. Pearson.
• Klebaner, F. C. (2012). Introduction to stochastic calculus with applications. 3rd Ed. Imperial College Press.
• Papoulis, A., & Pillai, S. U. (2002). Probability, random variables, and stochastic processes. 4th Ed. Tata McGraw-Hill Education.
• Gray, R. M. (2009). Probability, random processes, and ergodic properties. Springer Science & Business Media.
• Stirzaker, D. (2005). Stochastic processes and models. Oxford University Press.
• Grimmett, G., & Stirzaker, D. (2001). Probability and random processes. 3rd Ed. Oxford University Press.
• Kao, E. P. (1997). An introduction to stochastic processes. Duxbury.
• Ross, S. M. (1996). Stochastic processes. 2nd Ed. John Wiley & Sons.
• Stark, H., & Woods, J. W. (1994). Probability, random processes, and estimation theory for engineers. Prentice Hall.
• Helstrom, C.W., (1992). Probability and Stochastic Processes for Engineers. 2nd Ed. Addison-Wesley.
• Bartlett, M. S. (1978). An Introduction to Stochastic Processes, with Special Reference to Methods and Applications. Cambridge University Press.
• Doob, J. L. (1953).Stochastic Processes. Wiley.

Scholarly Articles

• Geering, H. P., Dondi, G., Herzog, F., & Keel, S. (2011). Stochastic systems. Course script. (link)

Software

• StochPy - Python
• Bridge.jl - Julia
• Brownian.jl - Julia
• RandLib - C++
• binary-martingale - Python

See also

• State Space Models / Linear Dynamical Systems
• Kalman filter
• Information Theory
• Probability Theory
• Estimation Theory / System Identification / Statistical Signal Processing
• Bayesian Nonparametrics
• Monte Carlo Methods / Stochastic Simulation
• Computational Finance
• Artificial Neural Networks

Other Resources

• Stochastic Processes - Notebook
• Random Fields - Notebook
• Time Series - Notebook