Skip to main content
SpringerLink
Log in

A Fluid Introduction to Brownian Motion and Stochastic Integration

  • Conference paper
  • First Online:
Book cover Matrix-Analytic Methods in Stochastic Models

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 27))

Abstract

This chapter provides an introduction to Brownian motion and stochastic integrals using linear fluid flows on finite state Markov chains. Many numerical examples are presented setting the stage for the development of algorithms for stochastic integration via the well-studied and easily understood fluid flow models driven by finite state Markov chains.

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

Access this chapter

Log in via an institution
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 EPUB and 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

References

  1. Ahn, S., Ramaswami, V.: Transient analysis of fluid flow models via stochastic coupling to a queue. Stoch. Models 20(1), 71–101 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahn, S., Ramaswami, V.: Transient analysis of fluid flow models via elementary level crossing argument. Stoch. Models 22(1), 129–148 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ahn, S., Ramaswami, V.: Efficient algorithms for transient analysis of stochastic fluid flow models. J. Appl. Probab. 42(2), 531–549 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ahn, S., Badescu, A.L, Ramaswami, V.: Transient analysis of finite buffer fluid flows and risk models with a dividend barrier. QUESTA 55, 207–222 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Asmussen, S: Stationary distributions of fluid flow models with or without Brownian noise. Stoch. Models 11, 1–20 (1995)

    Article  MathSciNet  Google Scholar 

  6. Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)

    Book  MATH  Google Scholar 

  7. Borodin, A.N., Salminen, P.: Handbook of Brownian Motion - Facts and Formulae. Birkhäuser, Basel, Boston and Berlin (1996)

    Book  MATH  Google Scholar 

  8. Breiman, L.: Probability. Addison-Wesley, Reading, MA (1968)

    MATH  Google Scholar 

  9. Chung, K.L.: A Course in Probability Theory. Harcourt, Brace & World (1968)

    MATH  Google Scholar 

  10. Ciesielski, Z.: Hölder conditions for realisations of Gaussian processes. Trans. Am. Math. Soc. 99, 403–413 (1961)

    MathSciNet  MATH  Google Scholar 

  11. Donsker, M.: An invariance principle for certain probability limit theorems. Mem. Am. Math. Soc. 6, 1–10 (1951)

    MathSciNet  Google Scholar 

  12. Higham, D.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Itô, K.: Stochastic Integral. Proc. Imper. Acad. Tokyo 20, 519–524 (1944)

    Article  MATH  Google Scholar 

  14. Jentzen, A., Kloeden, P.E.: Pathwise Taylor schemes for random ordinary differential equations. BIT Numer. Math. 49, 113–140 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Karandikar, R.: Pathwise solutions of stochastic differential equations. Sankhya A 43(2), 121–132 (1981)

    MathSciNet  MATH  Google Scholar 

  16. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1991)

    MATH  Google Scholar 

  17. Kloeden, P.E., Platten, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1995)

    Google Scholar 

  18. Latouche, G., Takine, T.: Markov renewal fluid queues. J. Appl. Probab. 41, 746–757 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nelson, D., Ramaswamy, K.: Simple binomial processes as diffusion approximations in financial models. Rev. Financ. Stud. 3, (3), 393–430 (1990)

    Google Scholar 

  20. Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin (2003)

    Google Scholar 

  21. Protter, P.: Stochastic Integration and Differential Equations. Springer, New York (1990)

    MATH  Google Scholar 

  22. Ramaswami, V.: Matrix analytic methods for stochastic fluid flows. In: Smith, D., Key, P. (eds.) Teletraffic Engineering in a Competitive World. Proceedings of the 16th International Teletraffic Congress, pp. 1019–1030. Elsevier, New York (1999)

    Google Scholar 

  23. Ramaswami, V.: Passage times in fluid models with application to risk processes. Meth. Comput. Appl. Probab. 8, 497–515 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales, vol. I. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  25. Shreve, S.E.: Stochastic Calculus for Finance, vol. 2. Continuous Time Models. Springer, Berlin (2004)

    Book  Google Scholar 

  26. Skorohod, A.V.: Limit theorems for stochastic processes. Theor. Probab. Appl. 1, 261–290 (1956)

    Article  MathSciNet  Google Scholar 

  27. Steele, M.J.: Ito calculus. In: Teugels, J., Sundt, B. (eds.) The Encyclopedia of Actuarial Science. Wiley, New York (2004)

    Google Scholar 

  28. Stratonovich, R.L.: A new representation for stochastic integrals and equations. J. SIAM Contr. 4(2), 362–371 (1966)

    Article  MathSciNet  Google Scholar 

  29. Szabados, T.: An elementary approach to the Wiener process and stochastic integrals. Stud. Sci. Math. Hung. 41, 101–126 (2004)

    MathSciNet  MATH  Google Scholar 

  30. Szabados, T., Székely, B.: Stochastic integration based on simple, symmetric random walks. J. Theor. Probab. 22, 203–219 (2009)

    Article  MATH  Google Scholar 

  31. Whitt, W.: Stochastic Process Limits. Springer, Berlin (2002)

    MATH  Google Scholar 

  32. Wiener, N.: Un problème de probabilités énombrables. Bull. Soc. Math. Fr. 52, 569–568 (1924)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. AT&T Labs, Research, 180 Park Avenue, Florham Park, NJ, 07932, USA

    Vaidyanathan Ramaswami

Authors
  1. Vaidyanathan Ramaswami
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vaidyanathan Ramaswami .

Editor information

Editors and Affiliations

  1. , Faculté des Sciences, Univ. Libre de Bruxelles, Blvd. du Triomphe, CP 212, Bruxelles, 1050, Belgium

    Guy Latouche

  2. At&T Labs Research, Park Avenue 180, Florham Park, 07932, New Jersey, USA

    Vaidyanathan Ramaswami

  3. , IEOR, Columbia University, West 120th Street 500, New York City, 10027, New York, USA

    Jay Sethuraman

  4. , Dept of Industrial Engineering and OR, Columbia University, 312 Mudd Bldg., New York, 10027, New York, USA

    Karl Sigman

  5. IBM Thomas J. Watson Research Center, Kitchawan Road 1101, Yorktown Heights, 10598, New York, USA

    Mark S. Squillante

  6. , IEOR, Columbia University, West 120th Street 500, New York City, 10027, New York, USA

    David D. Yao

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this paper

Cite this paper

Ramaswami, V. (2013). A Fluid Introduction to Brownian Motion and Stochastic Integration. In: Latouche, G., Ramaswami, V., Sethuraman, J., Sigman, K., Squillante, M., D. Yao, D. (eds) Matrix-Analytic Methods in Stochastic Models. Springer Proceedings in Mathematics & Statistics, vol 27. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4909-6_10

Download citation

Publish with us

Policies and ethics

Search

Navigation