Stochastic Processes

  • Peter Schuster
Part of the Springer Series in Synergetics book series (SSSYN)


Stochastic processes are defined and grouped into different classes, their basic properties are listed and compared. The Chapman–Kolmogorov equation is introduced, transformed into a differential version, and used to classify the three major types of processes: (i) drift and (ii) diffusion with continuous sample paths, and (iii) jump processes which are essentially discontinuous. In pure form these prototypes are described by Liouville equations, stochastic diffusion equations, and master equations, respectively. The most popular and most frequently used continuous equation is the Fokker–Planck (FP) equation, which describes the evolution of a probability density by drift and diffusion. The pendant to FP equations on the discontinuous side are master equations which deal only with jump processes and represent the appropriate tool for modeling processes described by discrete variables. For technical reasons they are often difficult to handle unless population sizes are relatively small. Particular emphasis is laid on modeling conventional and anomalous diffusion processes. Stochastic differential equations (SDEs) model processes at the level of random variables by solving ordinary differential equations upon which a diffusion process, called a Wiener process, is superimposed. Ensembles of individual trajectories of SDEs are equivalent to time dependent probability densities described by Fokker–Planck equations.


Chapman Kolmogorov Differential Equations Stochastic Differential Equations (SDE) Time-dependent Probability Density Wiener Process Master Equation 
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  1. 10.
    Anderson, B.D.O.: Reverse-time diffusion equation models. Stoch. Process. Appl. 12, 313–326 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 14.
    Anderson, P.W.: More is different. Broken symmetry and the nature of the hierarchical stucture of science. Science 177, 393–396 (1972)Google Scholar
  3. 18.
    Applebaum, D.: Lévy processes – From probability to finance and quantum groups. Not. Am. Math. Soc. 51, 1336–1347 (2004)MathSciNetzbMATHGoogle Scholar
  4. 21.
    Arfken, G.B., Weber, H.J.: Mathematical Methods for Physicists, fifth edn. Harcourt Academic Press, San Diego (2001)Google Scholar
  5. 22.
    Arnold, L.: Stochastic Differential Equations. Theory and Applications. Wiley, New York (1974)zbMATHGoogle Scholar
  6. 31.
    Bachelier, L.: Théorie de la spéculation. Annales scientifiques de l’É.N.S. 3e série 17, 21–86 (1900)Google Scholar
  7. 43.
    Bergström, H.: On some expansions of stable distribution functions. Ark. Math. 2, 375–378 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 53.
    Birkhoff, G.D.: Proof of the ergodic theorem. Proc. Natl. Acad. sci. USA 17, 656–660 (1931)ADSCrossRefzbMATHGoogle Scholar
  9. 61.
    Bouchaud, J.P., Georges, A.: Anomalous diffusion in disordered madia: Statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)ADSMathSciNetCrossRefGoogle Scholar
  10. 66.
    Brockmann, D., Hufnagel, L., Geisel, T.: The scaling laws of human travel. Nature 439, 462–465 (2006)ADSCrossRefGoogle Scholar
  11. 67.
    Brockwell, P.J., Davis, R.A.: Introduction to Time Series and Forecasting. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  12. 68.
    Brockwell, P.J., Davis, R.A., Yang, Y.: Continuous-time Gaussian autoregression. Stat. Sin. 17, 63–80 (2007)zbMATHGoogle Scholar
  13. 69.
    Brown, R.: A brief description of microscopical observations made in the months of June, July and August 1827, on the particles contained in the pollen of plants, and on the general existence of active molecules in organic and inorganic bodies. Phil. Mag. Ser. 2 4, 161–173 (1828). First Publication: The Edinburgh New Philosophical Journal. July-September 1828, pp. 358–371Google Scholar
  14. 81.
    Chechkin, A.V., Metzler, R., Klafter, J., Gonchar, V.Y.: Introduction to the theory of Lévy flights. In: R. Klages, G. Radons, I.M. Sokolov (eds.) Anomalous Transport: Foundations and Applications, chap. 5, pp. 129–162. Wiley-VCH Verlag GmbH, Weinheim, DE (2008)Google Scholar
  15. 91.
    Cox, D.R., Miller, H.D.: The Theory of Stochastic Processes. Methuen, London (1965)zbMATHGoogle Scholar
  16. 95.
    Crank, J.: The Mathematics of Diffusion. Clarendon Press, Oxford (1956)zbMATHGoogle Scholar
  17. 106.
    Devroye, L.: Non-Uniform Random Variate Generation. Springer, New York (1986)CrossRefzbMATHGoogle Scholar
  18. 119.
    Dyson, F.: A meeting with Enrico Fermi. How one intuitive physicist rescued a team from fruitless research. Nature 427, 297 (2004)Google Scholar
  19. 130.
    Eigen, M.: Selforganization of matter and the evolution of biological macromolecules. Naturwissenschaften 58, 465–523 (1971)ADSCrossRefGoogle Scholar
  20. 133.
    Einstein, A.: Über die von der molekular-kinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annal. Phys. (Leipzig) 17, 549–560 (1905)Google Scholar
  21. 135.
    Elliot, R.J., Anderson, B.D.O.: Reverse-time diffusions. Stoch. Process. Appl. 19, 327–339 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 136.
    Elliot, R.J., Kopp, A.E.: Mathematics of Financial Markets, 2nd edn. Springer, New York (2005)zbMATHGoogle Scholar
  23. 149.
    Farlow, S.J.: Partial Differential Equations for Scientists and Engineers. Dover Publications, New York (1982)zbMATHGoogle Scholar
  24. 150.
    Feigenbaum, M.J.: Universal behavior in nonlinear systems. Physica D 7, 16–39 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 158.
    Feller, W.: On the theory of stochastic processes, with particular reference to applications. In: The Regents of the University of California (ed.) Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, pp. 403–432. University of California Press, Berkeley (1949)Google Scholar
  26. 161.
    Feller, W.: An Introduction to Probability Theory and Its Application, vol. II, 2nd edn. Wiley, New York (1971)Google Scholar
  27. 178.
    Fisk, D.L.: Quasi-martingales. Trans. Am. Math. Soc. 120, 369–389 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 193.
    Gardiner, C.W.: Handbook of Stochastic Methods, 1st edn. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
  29. 194.
    Gardiner, C.W.: Stochastic Methods. A Handbook for the Natural Sciences and Social Sciences, fourth edn. Springer Series in Synergetics. Springer, Berlin (2009)Google Scholar
  30. 202.
    Gibbs, J.W.: Elementary Principles in Statistical Mechanics. Charles Scribner’s Sons, New York (1902). Reprinted 1981 by Ox Bow Press, Woodbridge, CTGoogle Scholar
  31. 205.
    Gihman, I.F., Skorohod, A.V.: The Theory of Stochastic Processes. Vol. I, II, and III. Springer, Berlin (1975)Google Scholar
  32. 208.
    Gillespie, D.T.: Markov Processes: An Introduction for Physical Scientists. Academic Press, San Diego (1992)zbMATHGoogle Scholar
  33. 210.
    Gillespie, D.T.: Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral. Phys. Rev. E 54, 2084–2091 (1996)ADSMathSciNetCrossRefGoogle Scholar
  34. 211.
    Gillespie, D.T.: The chemical Langevin equation. J. Chem. Phys. 113, 297–306 (2000)ADSCrossRefGoogle Scholar
  35. 213.
    Gillespie, D.T.: Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem. 58, 35–55 (2007)ADSCrossRefGoogle Scholar
  36. 214.
    Gillespie, D.T., Seitaridou, E.: Simple Brownian Diffusion. An Introduction to the Standard Theoretical Models. Oxford University Press, Oxford (2013)zbMATHGoogle Scholar
  37. 216.
    Goel, N.S., Richter-Dyn, N.: Stochastic Models in Biology. Academic Press, New York (1974)Google Scholar
  38. 218.
    Goychuk, I.: Viscoelastic subdiffusion: Generalized langevin equation approach. Adv. Chem. Phys. 150, 187–253 (2012)Google Scholar
  39. 231.
    Hamilton, J.D.: Time Series Analysis. Princeton University Press, Princeton (1994)zbMATHGoogle Scholar
  40. 232.
    Hamilton, W.R.: On a general method in dynamics. Philos. Trans. R. Soc. Lond. II for 1834, 247–308 (1834)Google Scholar
  41. 233.
    Hamilton, W.R.: Second essay on a general method in dynamics. Philos. Trans. R. Soc. London I for 1835, 95–144 (1835)Google Scholar
  42. 244.
    Haubold, H.J., Mathai, M.A., Saxena, R.K.: Mittag-Leffler functions and their applications. J. Appl. Math. 2011, e298,628 (2011). Hindawi Publ. Corp.Google Scholar
  43. 245.
    Haussmann, U.G., Pardoux, E.: Time reversal of diffusions. Ann. Probab. 14, 1188–1205 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 269.
    Humphries, N.E., Queiroz, N., Dyer, J.R.M., Pade, N.G., Musyl, M.K., Schaefer, K.M., Fuller, D.W., Brunnschweiler, J.M., Doyle, T.K., Houghton, J.D.R., Hays, G.C., Jones, C.S., Noble, L.R., Wearmouth, V.J., Southall, E.J., Sims, D.W.: Environmental context explains Lévy and Brwonian movement patterns of marine predators. Nature 465, 1066–1069 (2010)ADSCrossRefGoogle Scholar
  45. 272.
    Itō, K.: Stochastic integral. Proc. Imp. Acad. Tokyo 20, 519–524 (1944)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 273.
    Itō, K.: On stochastic differential equations. Mem. Am. Math. Soc. 4, 1–51 (1951)Google Scholar
  47. 277.
    Jacobs, K.: Stochastic processes for Physicists. Understanding Noisy Systems. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  48. 290.
    Karlin, S., Taylor, H.M.: A First Course in Stochastic Processes, 2nd edn. Academic Press, New York (1975)zbMATHGoogle Scholar
  49. 304.
    Kimura, M.: The Neutral Theory of Molecular Evolution. Cambridge University Press, Cambridge (1983)CrossRefGoogle Scholar
  50. 325.
    Langevin, P.: Sur la théorie du mouvement Brownien. Comptes Rendues hebdomadaires des Séances de L’Academié des Sceinces 146, 530–533 (1908)zbMATHGoogle Scholar
  51. 340.
    Lewis, W.C.M.: Studies in catalysis. Part IX. The calculation in absolute measure of velocity constants and equilibrium constants in gaseous systems. J. Chem. Soc. Trans. 113, 471–492 (1918)Google Scholar
  52. 352.
    Liouville, J.: Note sur la théorie de la variation des constantes arbitraires. Journal de Mathématiques pure et appliquées 3, 342–349 (1838). In French.Google Scholar
  53. 353.
    Liouville, J.: Mémoire sur l’intégration des équations différentielles du mouvement quelconque de points matériels. Journal de Mathématiques pure et appliquées 14, 257–299 (1849). In French.Google Scholar
  54. 364.
    Mahnke, R., Kaupužs, J., Lubashevsky, I.: Physics of Stochastic Processes. How Randomness Acts in Time. Wiley-VCh Verlag, Weinheim (Bergstraße), DE (2009)zbMATHGoogle Scholar
  55. 366.
    Mandelbrot, B.B.: The Fractal Geometry of Nature, updated edn. W. H. Freeman Company, New York (1983)Google Scholar
  56. 367.
    Mansuy, R.: The origins of the word “martingale”. Electron. J. Hist. Probab. Stat. 5 (1), 1–10 (2009). Translated by Ronald Sverdlove from the French Histoire des martigales. Mathématiques Sciences Humaines 43 (169), 105–113 (2005)Google Scholar
  57. 374.
    Mathai, A.M., Saxena, R.K., Haubold, H.J.: A certain class of Laplace transforms with applications to reaction and reaction-diffusion equations. Astrophys. Space Sci. 305, 283–288 (2006)ADSCrossRefzbMATHGoogle Scholar
  58. 388.
    Medvegyev, P.: Stochastic Integration Theory. Oxford University Press, New York (2007)zbMATHGoogle Scholar
  59. 396.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. 399.
    Mittag-Leffler, M.G.: Sur la nouvelle fonction E α(x). C. R. Acad. Sci. Paris Ser. II 137, 554–558 (1903)zbMATHGoogle Scholar
  61. 407.
    Montroll, E.W., Weiss, G.H.: Random walks on lattices. II. J. Math. Phys. 6, 167–181 (1965)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. 408.
    Moore, C.C.: Ergodic theorem, ergodic theory and statistical mechanics. Proc. Natl. Acad. Sci. USA 112, 1907–1911 (2015)ADSMathSciNetCrossRefGoogle Scholar
  63. 415.
    Moyal, J.E.: Stochastic processes and statistical physics. J. R. Stat. Soc. B 11, 150–210 (1949)MathSciNetzbMATHGoogle Scholar
  64. 431.
    Øksendal, B.K.: Stochastic Differential Equations. An Introduction with Applications, 6th edn. Springer, Berlin (2003)Google Scholar
  65. 436.
    Papapantoleon, A.: An Introduction to Lévy Processes with Applications in Finance. arXiv, Princeton, NJ (2008). ArXiv:0804.0482v2 retrieved July 27, 2015Google Scholar
  66. 444.
    Pearson, K.: On the criterion that a given system of deviations form the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philos. Mag. Ser. 5 50 (302), 157–175 (1900)Google Scholar
  67. 450.
    Philibert, J.: One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals 4, 6.1–6.19 (2006)Google Scholar
  68. 454.
    Pollard, H.: The representatioin of \(e^{-x^{\lambda } }\) as a Laplace intgeral. Bull. Am. Math. Soc. 52, 908–910 (1946)MathSciNetCrossRefzbMATHGoogle Scholar
  69. 460.
    Protter, P.E.: Stochastic Intergration and Differential Equations, Applications of Mathematics, vol. 21, 2nd edn. Springer, Berlin (2004)Google Scholar
  70. 467.
    Riley, K.F., Hobson, M.P., Bence, S.J.: Mathematical Methods for Physics and Engineering, 2nd edn. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  71. 468.
    Risken, H.: TheFokker-Planck Equation. Methods of Solution and Applications, 2nd edn. Springer, Berlin (1989)Google Scholar
  72. 475.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions, 2nd edn. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
  73. 477.
    Scher, H., Shlesinger, M.F., Bendler, J.T.: Time scale invariance in transport and relaxation. Phys. Today 44 (1), 26–34 (1991)ADSCrossRefGoogle Scholar
  74. 480.
    Schoutens, W.: Lévy Processes in Finance. Wiley Series in Probability and Statistics. Wiley, Chichester (2003)CrossRefGoogle Scholar
  75. 500.
    Sharpe, M.J.: Transformations of diffusion by time reversal. Ann. Probab. 8, 1157–1162 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  76. 503.
    Sotiropoulos, V., Kaznessis, Y.N.: Analytical derivation of moment equations in stochastic chemical kinetics. Chem. Eng. Sci. 66, 268–277 (2011)CrossRefGoogle Scholar
  77. 506.
    Stepanow, S., Schütz, G.M.: The distribution function os a semiflexible polymer and random walks with constraints. Europhys. Lett. 60, 546–551 (2002)ADSCrossRefGoogle Scholar
  78. 512.
    Stratonovich, R.L.: Introduction to the Theory of Random Noise. Gordon and Breach, New York (1963)zbMATHGoogle Scholar
  79. 531.
    Tolman, R.C.: The Principle of Statistical Mechanics. Oxford University Press, Oxford (1938)zbMATHGoogle Scholar
  80. 534.
    Uhlenbeck, G.E., Ornstein, L.S.: On the theory of the Brownian motion. Phys. Rev. 36, 823–841 (1930)ADSCrossRefzbMATHGoogle Scholar
  81. 535.
    Ullah, M., Wolkenhauer, O.: Family tree of Markov models in systems biology. IET Syst. Biol. 1, 247–254 (2007)CrossRefGoogle Scholar
  82. 536.
    Ullah, M., Wolkenhauer, O.: Stochastic Approaches for Systems Biology. Springer, New York (2011)CrossRefzbMATHGoogle Scholar
  83. 537.
    van den Berg, T.: Calibrating the Ornstein-Uhlenbeck-Vasicek model. Sitmo – Custom Financial Research and Development Services, (2011). Retrieved April 20, 2014
  84. 541.
    van Kampen, N.G.: The expansion of the master equation. Adv. Chem. Phys. 34, 245–309 (1976)Google Scholar
  85. 542.
    van Kampen, N.G.: Remarks on non-markov processes. Braz. J. Phys. 28, 90–96 (1998)ADSGoogle Scholar
  86. 543.
    van Kampen, N.G.: Stochastic Processes in Physics and Chemistry, 3rd edn. Elsevier, Amsterdam (2007)zbMATHGoogle Scholar
  87. 546.
    Vasicek, O.: An equlibrium characterization of the term structure. J. Financ. Econ. 5, 177–188 (1977)CrossRefGoogle Scholar
  88. 551.
    Viswanathan, G.M., Raposo, E.P., da Luz, M.G.E.: Lévy flights and superdiffusion in the context of biological encounters and random searches. Phys. Life Rev. 5, 133–150 (2008)ADSCrossRefGoogle Scholar
  89. 558.
    von Neumann, J.: Proof of the quasi-ergodic hypothesis. Proc. Natl. Acad. Sci. USA 4, 70–82 (1932)ADSCrossRefzbMATHGoogle Scholar
  90. 559.
    von Smoluchowski, M.: Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Ann. Phys. (Leipzig) 21, 756–780 (1906)Google Scholar
  91. 564.
    Wegscheider, R.: Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reaktionskinetik homogener Systeme. Mh. Chem. 32, 849–906 (1911). In GermanzbMATHGoogle Scholar
  92. 565.
    Wei, W.W.S.: Time Series Analysis. Univariate and Multivariate Methods. Addison-Wesley Publishing, Redwood City (1990)Google Scholar
  93. 567.
    Weisstein, E.W.: Cross-Correlation. MathWorld - A Wolfram Web Resource. The Wolfram Centre, Long Hanborough, UK., retrieved July 17, 2015
  94. 568.
    Weisstein, E.W.: Fourier Transform. MathWorld - A Wolfram Web Resource. The Wolfram Centre, Long Hanborough, UK., retrieved July 17, 2015
  95. 574.
    Williams, D.: Diffusions, Markov Processes and Martingales. Volume 1: Foundations. Wiley, Chichester (1979)Google Scholar
  96. 578.
    Wold, H.: A Study in the Analysis of Time Series, second revised edn. Almqvist and Wiksell Book Co., Uppsala, SE (1954). With an appendix on Recent Developments in Time Series Analysis by Peter WhittleGoogle Scholar
  97. 580.
    Wright, S.: The roles of mutation, inbreeding, crossbreeding and selection in evolution. In: Jones, D.F. (ed.) Int. Proceedings of the Sixth International Congress on Genetics, vol. 1, pp. 356–366. Brooklyn Botanic Garden, Ithaca (1932)Google Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Peter Schuster
    • 1
  1. 1.Institut für Theoretische ChemieUniversität WienWienAustria

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