Stochastic Differential Equations

  • Gregory S. Chirikjian
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


The chapter begins with Section 4.1 in which motivational examples of random walks and stochastic phenomena in nature are presented. In Section 4.2 the concept of random processes is introduced in a more precise way. In Section 4.3 the concept of a Gaussian and Markov random process is developed. In Section 4.4 the important special case of white noise is defined. White noise is the driving force for all of the stochastic processes studied in this book. Other sections in this chapter define Itô and Stratonovich stochastic differential equations (SDEs), their properties and corresponding Fokker–Planck equations, which describe how probability densities associated with SDEs evolve over time. In particular, Section 4.7 examines the Fokker–Planck equation for a particular kind of SDE called an Ornstein–Uhlenbeck process. And Section 4.8 examines how SDEs and Fokker–Planck equations change their appearance when different coordinate systems are used. The main points that the reader should take away from this chapter are: Whereas a deterministic system of ordinary differential equations that satisfies certain conditions (i.e., the Lipschitz conditions) are guaranteed to have a unique solution for any given initial conditions, when random noise is introduced the resulting “stochastic differential equation” will not produce repeatable solutions. It is the ensemble behavior of the sample paths obtained from numerically solving a stochastic differential equation many times that is important. This ensemble behavior can be described either as a stochastic integral (of which there are two main types, called Itˆo and Stratonovich), or by using a partial differential equation akin to the diffusion equations studied in Chapter 2, which is called the Fokker–Planck (or forward Kolmogorov) equation. Two different forms of the Fokker–Planck equation exist, corresponding to the interpretation of the solution of a given SDE as being either an Itˆo or Stratonovich integral, and an analytical apparatus exists for converting between these forms. Multi-dimensional SDEs in Rn can be written in Cartesian or curvilinear coordinates, but care must be taken when converting between coordinate systems because the usual rules of multivariable calculus do not apply in some situations.


Brownian Motion Sample Path Wiener Process Planck Equation Coordinate Change 
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  1. 1.
    Bouleau, N., Lépingle, D., Numerical Methods for Stochastic Processes, John Wiley & Sons, New York, 1994.MATHGoogle Scholar
  2. 2.
    Casimir, H.B.G., “On onsager's principle of microscopic reversibility,” Rev. Mod. Phys., 17(2–3), pp. 343–350, 1945.CrossRefGoogle Scholar
  3. 3.
    Chirikjian, G.S., Kyatkin, A.B., Engineering Applications of Noncommutative Harmonic Analysis, CRC Press, Boca Raton, FL, 2001.MATHGoogle Scholar
  4. 4.
    Doob, J.L., Stochastic Processes, John Wiley & Sons, New York, 1953.MATHGoogle Scholar
  5. 5.
    Durrett, R., Brownian Motion and Martingales in Analysis, Wadsworth, Belmont, CA, 1984.MATHGoogle Scholar
  6. 6.
    Einstein, A., Investigations on the Theory of the Brownian Movement, Dover, New York, 1956.MATHGoogle Scholar
  7. 7.
    Fokker, A.D., “Die Mittlere Energie rotierender elektrischer Dipole in Strahlungs Feld,” Ann. Phys., 43, pp. 810–820, 1914.CrossRefGoogle Scholar
  8. 8.
    Fowler, R.H., Statistical Mechanics, Cambridge University Press, London, 1929.MATHGoogle Scholar
  9. 9.
    Fowler, R.H., Philos. Mag., 47, p. 264, 1924.Google Scholar
  10. 10.
    Gard, T.C., Introduction to Stochastic Differential Equations, Marcel Dekker, New York, 1988.MATHGoogle Scholar
  11. 11.
    Gardiner, C.W., Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences, 3rd ed., Springer-Verlag, Berlin, 2004.Google Scholar
  12. 12.
    Higham, D.J., “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Rev., 43, pp. 525–546, 2001.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Itô, K., McKean, H.P., Jr. Diffusion Processes and their Sample Paths, Springer-Verlag, Berlin, 1974.MATHGoogle Scholar
  14. 14.
    Karatzas, I., Shreve, S.E., Brownian Motion and Stochastic Calculus, 2nd ed., Springer, New York, 1991.MATHGoogle Scholar
  15. 15.
    Karlin, S., Taylor, H.M., An Introduction to Stochastic Modeling, 3rd ed., Academic Press, San Diego, 1998.MATHGoogle Scholar
  16. 16.
    Kloedon, P.E., Platen, E., Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.Google Scholar
  17. 17.
    Knight, F.B., Essentials of Brownian Motion, Math. Survey 18, American Mathematical Society, Providence, RI, 1981.MATHGoogle Scholar
  18. 18.
    Kolmogorov, A.N., “Uber die analytischen Methoden in der Wahrscheinlichkeitsrechnung,” Math. Ann., 104, pp. 415–458, 1931.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kuo, H.-H., Introduction to Stochastic Integration, Springer, New York, 2006.MATHGoogle Scholar
  20. 20.
    Langevin, P., “Sur la théorie du mouvement brownien,” C. R. Acad. Sci. Paris, 146, pp. 530–533, 1908.MATHGoogle Scholar
  21. 21.
    Lévy, P., Processsus stochastiques et mouvement brownien, Gauthiers-Villars, Paris, 1948 (and 1965).Google Scholar
  22. 22.
    Maruyama, G., “Continuous Markov processes and stochastic equations,” Rend. Circ. Mat. Palermo, 4, pp. 48–90, 1955.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    McKean, H.P., Jr., Stochastic Integrals, Academic Press, New York, 1969.MATHGoogle Scholar
  24. 24.
    McShane, E.J., Stochastic Calculus and Stochastic Models, Academic Press, New York, 1974.MATHGoogle Scholar
  25. 25.
    Millstein, G.N., “A method of second order accuracy of stochastic differential equations,” Theory of Probability and Its Applications (USSR), 23, pp. 396–401, 1976.CrossRefGoogle Scholar
  26. 26.
    Millstein, G.N., Tretyakov, M.V., Stochastic Numerics for Mathematical Physics, Springer-Verlag, Berlin, 2004.Google Scholar
  27. 27.
    Øksendal, B., Stochastic Differential Equations, An Introduction with Applications, 5th ed., Springer, Berlin, 1998.Google Scholar
  28. 28.
    Onsager, L., “Reciprocal relations in irreversible processes, I, II,” Phys. Rev., 37, pp. 405–426, 38, pp. 2265–2280, 1931.MATHCrossRefGoogle Scholar
  29. 29.
    Planck, M., “Uber einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie,” Sitzungsber. Berlin Akad. Wiss., pp. 324–341, 1917.Google Scholar
  30. 30.
    Protter, P., Stochastic Integration and Differential Equations, Springer, Berlin, 1990.MATHGoogle Scholar
  31. 31.
    Rényi, A., Probability Theory, North-Holland, Amsterdam, 1970.Google Scholar
  32. 32.
    Ripley, B.D., Stochastic Simulation, John Wiley & Sons, New York, 1987.MATHCrossRefGoogle Scholar
  33. 33.
    Risken, H., The Fokker-Planck Equation, Methods of Solution and Applications, 2nd ed., Springer-Verlag, Berlin, 1989.MATHGoogle Scholar
  34. 34.
    Rogers, L.C.G., Williams, D., Diffusion, Markov Processes, and Martingales, Vols. 1 and 2, John Wiley & Sons, New York, 1987.Google Scholar
  35. 35.
    Stratonovich, R.L., Topics in the Theory of Random Noise, Vols. I and II, (translated by R.A. Silverman), Gordon and Breach, New York, 1963.Google Scholar
  36. 36.
    Stroock, D., Varadhan, S.R.S., Multidimensional Diffusion Processes, Grundlehren Series #233, Springer-Verlag, Berlin, 1979 (and 1998).MATHGoogle Scholar
  37. 37.
    Uhlenbeck, G.E., Ornstein, L.S., “On the theory of Brownian motion,” Phys. Rev., 36, pp. 823–841, 1930.CrossRefGoogle Scholar
  38. 38.
    van Kampen, N.G., Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1981.MATHGoogle Scholar
  39. 39.
    van Kampen, N.G., “Derivation of the phenomenological equations from the master equation: I. Even variables only; II. Even and odd variables,” Physica, 23, pp. 707–719, pp. 816–824, 1957.MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Wang, M.C., Uhlenbeck, G.E., “On the theory of Brownian motion II,” Rev. Mod. Phys., 7, pp. 323–342, 1945.CrossRefMathSciNetGoogle Scholar
  41. 41.
    Watanabe, S., Stochastic Differential Equations and Malliavin Calculus, Tata Institute, 1984.Google Scholar
  42. 42.
    Wiener, N., “Differential space,” J. Math. Phys., 2, pp. 131–174, 1923.Google Scholar

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© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringThe Johns Hopkins UniversityBaltimoreUSA

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