Skip to main content
SpringerLink
Log in
Book cover

From Stochastic Calculus to Mathematical Finance pp 227–248Cite as

On Existence and Uniqueness of Reflected Solutions of Stochastic Equations Driven by Symmetric Stable Processes

  • Chapter

This is a preview of subscription content, 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   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   54.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

 References

  1. Bertoin, J.: Lévy Processes. Cambridge University Press, 1996

    Google Scholar 

  2. Blumenthal, R. M., Getoor, R. K.: Markov Processes and Potential Theory. Academic Press, New York, 1968

    MATH  Google Scholar 

  3. Boylan, E. S.: Local times for a class of Markov processes. Illinois J. Math. 8, pp. 19-39 (1964)

    MATH  MathSciNet  Google Scholar 

  4. Chung, K. L., Williams, R. J.: Introduction to Stochastic Integration. Birkhäuser Verlag, 1983

    Google Scholar 

  5. Engelbert, H. J., Kurenok, V. P.: On one-dimensional stochastic equations driven by symmetric stable processes. in: Buckdahn, R., Engelbert, H. J., Yor, M. (eds), Stochastic Processes and Related Topics, pp. 81-110, Taylor and Francis Group, 2002

    Google Scholar 

  6. Jacod, J.: Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Math., Vol. 714, Springer-Verlag, Berlin, 1979

    Google Scholar 

  7. Jacod, J., Shiryaev, A.: Limit theorems for stochastic processes. 2nd ed., Grundlehren der Mathematischen Wissenschaften, Vol. 288, Springer-Verlag, Berlin, 2003

    Google Scholar 

  8. Kallenberg, O.: Foundations of Modern Probability. Springer Verlag, 1998

    Google Scholar 

  9. Knight, F. B.: The local time at zero of the reflected symmetric stable process. Z. Wahrscheinlichkeitstheorie verw. Geb. 19, pp. 180-190 (1971)

    Article  MathSciNet  Google Scholar 

  10. Pragarauskas, H., Zanzotto, P. A.: On one-dimensional stochastic differential equations driven by stable processes. Liet. Mat. Rink., Vol. 40, N 1, pp. 1-24 (2000)

    MathSciNet  Google Scholar 

  11. Ray, D.: Stable processes with an absorbing barrier. Trans. Am. Math. Soc. 87, pp. 187-197 (1958)

    Article  Google Scholar 

  12. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer Verlag, 1994

    Google Scholar 

  13. Rosinski, J., Woyczynski, W.: On Itô stochastic integration with respect to p-stable motion: inner clock, integrability of sample paths, double and multiple integrals. Ann. Probab., Vol. 14, N 1, pp. 271-286 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  14. Rozkosz, A., Slominski, L.: On stability and existence of solutions of SDEs with reflection an the boundary. Stoch. Process. Appl. 68, pp. 285-302 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Uni-versity Press, 1999

    Google Scholar 

  16. Schmidt, W. M.: On stochastic differential equations with reflecting barriers. Math. Nachr., Vol. 142, pp. 135-148 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  17. Skorohod, A. V.: Stochastic equations for diffusion processes in a bounded re-gion. Theory Probab. Appl. 6, pp. 264-274 (1961)

    Google Scholar 

  18. Skorohod, A. V.: Random Processes with Independent Increments. Kluwer, Dor- drecht, Netherlands, 1991

    MATH  Google Scholar 

  19. Slominski, L.: On existence, uniqueness and stability of solutions of multidimen-sional SDE’s with reflecting boundary condition. Inst. Henri Poincaré 29, N 2, pp. 169-198 (1993)

    MathSciNet  Google Scholar 

  20. Slominski, L.: On approximation of solutions of multidimensional SDE’s with reflecting boundary conditions. Stoch. Process. Appl. 50, pp. 197-219 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  21. Tanaka, H.: Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9, pp. 163-177 (1978)

    Google Scholar 

  22. Watanabe, S.: On stable processes with boundary conditions, J. Math. Soc. Japan, Vol. 14, N 2, pp. 170-198 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  23. Zanzotto, P. A.: On solutions of one-dimensional stochastic differential equations driven by stable Lévy motion. Stoch. Process. Appl. 68, pp. 209-228 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. Zanzotto, P. A.: On stochastic differential equations driven by a Cauchy process and other stable Lévy motions. Ann. Probab. 30, N 2, pp. 802-825 (2002)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Stochastik, Friedrich-Schiller-Universität, Ernst-Abbe Platz 1–4, D-07743, Jena, Germany

    Hans-Jürgen Engelbert

  2. Department of Natural and Applied Sciences, University of Wisconsin-Green Bay, 2420 Nicolet Drive, 54311-7001, Green Bay, WI, USA

    Vladimir P. Kurenok

  3. Institut für Stochastik, Friedrich-Schiller-Universität, Ernst-Abbe Platz 1–4, D-07743, Jena, Germany

    Adrian Zalinescu

Authors
  1. Hans-Jürgen Engelbert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vladimir P. Kurenok
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Adrian Zalinescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Engelbert, HJ., Kurenok, V.P., Zalinescu, A. (2006). On Existence and Uniqueness of Reflected Solutions of Stochastic Equations Driven by Symmetric Stable Processes. In: From Stochastic Calculus to Mathematical Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30788-4_11

Download citation

Publish with us

Policies and ethics

Search

Navigation