Skip to main content

Part of the book series: Applications of Mathematics ((SMAP,volume 23))

  • 8873 Accesses

Abstract

In this chapter we shall describe several methods which allow a reduction in the variance of functionals of weak approximations of Ito diffusions. One method changes the underlying probability measure by means of a Girsanov transformation, another uses general principles of Monte-Carlo integration. Unbiased estimators are also constructed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographical Notes

  • Hammersley, J. M. and D. C. Handscomb (1964). Monte Carlo Methods. Methuen, London.

    Book  MATH  Google Scholar 

  • Karlin, S. and H. M. Taylor (1975). A First Course in Stochastic Processes ( 2nd ed. ). Academic Press, New York.

    MATH  Google Scholar 

  • Kloeden, P. E. and R. A. Pearson (1977). The numerical solution of stochastic differential equations. J. Austral. Math. Soc. Ser. B 20, 8–12.

    Article  MathSciNet  MATH  Google Scholar 

  • Lanska, V. (1979). Minimum contrast estimation in diffusion processes. J. AppL Probab. 16, 65–75.

    Article  MathSciNet  MATH  Google Scholar 

  • Le Gland, F. (1981). Estimation de paramétres dans les processus stochastiques en observation incompléte: Application d un probléme de radio-astronomie. Ph. D. thesis, Dr. Ing. Thesis, Univ. Paris IX (Dauphine).

    Google Scholar 

  • Liske, H. (1982). Distribution of a functional of a Wiener process. Theory of Random Processes 10, 50–54. (in Russian).

    MathSciNet  MATH  Google Scholar 

  • Ripley, B. D. (1983b). Stochastic Simulation. Wiley, New York.

    Google Scholar 

  • Schöner, G., H. Haken, and J. A. S. Kelso (1986). A stochastic theory of phase transitions in human hand movement. Biol. Cybern. 53, 247–257.

    Article  MATH  Google Scholar 

  • Bratley, P., B. L. Fox, and L. Schrage (1987). A Guide to Simulation (2nd ed.). Springer.

    Google Scholar 

  • Chang, C. C. (1987). Numerical solution of stochastic differential equations with constant diffusion coefficients. Math. Comp. 49, 523–542.

    Article  MathSciNet  MATH  Google Scholar 

  • Wagner, W. (1987a). Unbiased Monte-Carlo evaluation of certain functional integrals. J. Comput. Phys. 71 (1), 21–33.

    Article  MathSciNet  MATH  Google Scholar 

  • Wagner, W. (1988a). Monte-Carlo evaluation of functionals of solutions of stochastic differential equations. Variance reduction and numerical examples. Stochastic Anal. Appl. 6, 447–468.

    Article  MathSciNet  MATH  Google Scholar 

  • Milstein, G. N. (1988b). A theorem of the order of convergence of mean square approximations of systems of stochastic differential equations. Theory Probab. AppL 32, 738–741.

    Article  Google Scholar 

  • Wagner, W. (1989a). Stochastische numerische Verfahren zur Berechnung von Funktionalintegralen. (in German), Habilitation, Report 02/89, IMATH, Berlin.

    Google Scholar 

  • Wagner, W. (1989b). Unbiased Monte-Carlo estimators for functionals of weak solutions of stochastic differential equations. Stochastics Stochastics Rep. 28(1), 1–20.

    Google Scholar 

  • Wozniakowski, H. (1991). Average case complexity of multivariate integration. Bull. Austral. Math. Soc. 24 (1), 185–194.

    Article  MathSciNet  MATH  Google Scholar 

  • Yen, V. V. (1992). A stochastic Taylor formula for two-parameter stochastic processes. Probab. Math. Statist. 13 (1), 149–155.

    MathSciNet  MATH  Google Scholar 

  • Chorin, A. J. (1971). Hermite expansions in Monte-Carlo computation. J. Comput. Phys. 8, 472–482.

    Article  MathSciNet  MATH  Google Scholar 

  • Chorin, A. J. (1973a). Accurate evaluation of Wiener integrals. Math. Comp. 27, 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  • Chorin, A. J. (1973b). Numerical study of slightly viscous flow. J. Fluid Mech. 57, 785–786.

    Article  MathSciNet  Google Scholar 

  • Maltz, F. H. and D. L. Hitzl (1979). Variance reduction in Monte-Carlo computations using multi-dimensional Hermite polynomials. J. Comput. Phys. 32, 345–376.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Kloeden, P.E., Platen, E. (1992). Variance Reduction Methods. In: Numerical Solution of Stochastic Differential Equations. Applications of Mathematics, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-12616-5_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-12616-5_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08107-1

  • Online ISBN: 978-3-662-12616-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics