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Time-Varying Probabilities

  • Alexandre J. Chorin
  • Ole H. Hald
Chapter
Part of the Texts in Applied Mathematics book series (TAM, volume 58)

Abstract

There are many situations in which one needs to consider differential equations that contain a stochastic element, for example, equations in which the value of some coefficient depends on a measurement. The solution of the equation is then a function of the independent variables in the equation as well as of a point ω in some probability space; i.e., it is a stochastic process.

Keywords

Brownian Motion Stochastic Differential Equation Langevin Equation Gaussian Random Variable Planck Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

5.5. Bibliography

  1. [1].
    L. Arnold, Stochastic Differential Equations, Wiley, New York, 1973.Google Scholar
  2. [2].
    A.J. Chorin and X. Tu, Implicit sampling for particle filters, Proc. Nat. Acad. Sc. USA 106 (2009), pp. 17249–17254.Google Scholar
  3. [3].
    A.J. Chorin, M. Morzfeld and X. Tu, Implicit filters for data assimilation, Comm. Appl. Math. Comp. Sc. 5 (2010), pp. 221–240.Google Scholar
  4. [4].
    A. Doucet, S. Godsill, and C. Andrieu, On sequential Monte Carlo methods for Bayesian filtering, J. Stat. Comp. 10 (2000), pp. 197–208.Google Scholar
  5. [5].
    A. Doucet, N. de Freitas, and N. Gordon, Sequential Monte Carlo Methods in Practice, Springer, New York, 2001.Google Scholar
  6. [6].
    S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943), pp. 1–88; reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes, Dover, New York, 1954.Google Scholar
  7. [7].
    A.J. Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), pp. 785–796.Google Scholar
  8. [8].
    A.J. Chorin, Vortex methods, in Les Houches Summer School of Theoretical Physics, 59, (1995), pp. 67–109.Google Scholar
  9. [9].
    C.W. Gardiner, Handbook of Stochastic Methods, Springer, New York, 1985.Google Scholar
  10. [10].
    P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.Google Scholar
  11. [11].
    S. Neftci, An Introduction to the Mathematics of Financial Derivatives, Academic, New York, 2000.Google Scholar
  12. [12].
    B. Oksendal, Stochastic Differential Equations, Springer, New York, 1991.Google Scholar
  13. [13].
    J. Weare, Particle filters with path sampling and an application to a bimodal ocean current model. J. Comput. Phys. 228 (2009), pp. 4312–4333.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Alexandre J. Chorin
    • 1
  • Ole H. Hald
    • 2
  1. 1.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA
  2. 2.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA

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