Skip to main content

Path-space measure for stochastic differential equation with a coefficient of polynomial growth

  • Chapter
The Strength of Nonstandard Analysis
  • 1176 Accesses

Abstract

A σ-additive measure over a space of paths is constructed to give the solution to the Fokker-Planck equation associated with a stochastic differential equation with coefficient function of polynomial growth by making use of nonstandard analysis.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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.

References

  1. H.P. McKean, Jr., Stochastic Integrals, Academic Press, New York and London, 1969, (§3.6).

    MATH  Google Scholar 

  2. P. A. Loeb, “Conversion from nonstandard to standard measure spaces and applications in probability theory”, Trans. Amer. Math. Soc., 211 (1975) 113–122.

    Article  MATH  MathSciNet  Google Scholar 

  3. K. D. Stroyan and J. M. Bayod, Foundations of infinitesimal stochastic analysis, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1986.

    MATH  Google Scholar 

  4. I. V. Gigrsanov, “On transforming a certain class of stochastic processes by absolutely continuous substitution of measures”, Theor. Prob. Appl., 5 (1960) 285–301.

    Article  Google Scholar 

  5. T. Nakamura “A nonstandard representation of Feynman’s path integrals”, J. Math. Phys., 32 (1991) 457–463.

    Article  MATH  MathSciNet  Google Scholar 

  6. T. Nakamura, “Path space measures for Dirac and Schrödinger equations: Nonstandard analytical approach”, J. Math. Phys., 38 (1997) 4052–4072.

    Article  MATH  MathSciNet  Google Scholar 

  7. T. Nakamura, “Path space measure for the 3+1-dimensional Dirac equation in momentum space”, J. Math. Phys., 41 (2000) 5209–5222.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Wien

About this chapter

Cite this chapter

Nakamura, T. (2007). Path-space measure for stochastic differential equation with a coefficient of polynomial growth. In: van den Berg, I., Neves, V. (eds) The Strength of Nonstandard Analysis. Springer, Vienna. https://doi.org/10.1007/978-3-211-49905-4_21

Download citation

Publish with us

Policies and ethics