Skip to main content
SpringerLink
Log in

On the Pathwise Uniqueness of Solutions of One-Dimensional Stochastic Differential Equations with Jumps

  • Conference paper
  • First Online:
Statistical Methods and Applications in Insurance and Finance (CIMPA School 2013)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 158))

Included in the following conference series:

Abstract

We consider one-dimensional stochastic differential equations with jumps in the general case. We introduce new technics based on local time and we prove new results on pathwise uniqueness and comparison theorems. Our approach are very easy to handled. Similar equations without jumps were studied in the same context by [10, 15] and others authors.

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. Barndorff-Nielsen, O.E.: Processes of normal inverse Gaussian type. Finance Stochast. 2, 41–68 (1998)

    Google Scholar 

  2. Barlow, M.T., Perkins, E.: SDE’s singular increasing process. Stochastics 12, 229–249 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bass, R.: Stochastic differential equations driven by symmetric stable processes. In: Séminaire de probabilités, XXXVI, Lecture Notes in Math., Vol. 1801, pp. 302–313. Springer, Berlin (2003)

    Google Scholar 

  4. Bass, R.: Stochastic differential equations with jumps. Probab. Surv. 1, 1–19 (2004)

    Google Scholar 

  5. Cont, R., Tankov, P.: Financial modeling with jump processes, Chapman and Hall/CRC (2003)

    Google Scholar 

  6. Dawson, D.A., Li, Z.: Stochastic equations, flows and measure-valued processes. Ann. Probab. 40(2), 813–857 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Engelbert, H.J., Schmidt, W.: Strong Markov continous local martingales and solutions of one-Dimensional Stochastic differential equations. Part I. Math. Nachr. 143, 167–184 (1989)

    Article  MATH  Google Scholar 

  8. Fu, Z., Li, Z.: Stochastic equations of non-negative processes with jumps. Stochastic Process. Appl. 120(3), 306–330 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. North-Holland Mathematical Library, vol. 24, 2nd edn. North-Holland Publishing Co., Amsterdam (1989)

    Google Scholar 

  10. Le Gall, J. F.: One-dimensional stochastic differential equations involving the local cfftimes of unknown processes. In: Proceedings Stochastic Analysis and Applications (Swansea, 1983), Lecture Notes in Math., Vol. 1095, pp. 51–82. Springer, Berlin (1984)

    Google Scholar 

  11. Ornstein, L.S., Uhlenbeck, G.E.: On the theory of Brownain motion. Physical Review 36, 823–841 (1930)

    Article  MATH  Google Scholar 

  12. Ouknine, Y.: Généralisation d’un lemme de S. Nakao et applications. Stochastics 23(2), 149–157 (1988)

    Google Scholar 

  13. Ouknine, Y.: Fonctions de semimartingales et applications aux équations différentielles stochastiques. Stochast. Stochast. Rep. 28, 115–122 (1989)

    Article  MATH  Google Scholar 

  14. Ouknine, Y.: Quelques identités sur les temps locaux et unicité des solutions d’équations differentielles stochastiques avec reflection. Stochastic Process. Appl. 48(2), 335–340 (1993)

    Article  MathSciNet  Google Scholar 

  15. Ouknine, Y.: Unicité trajectorielle des equations différentielles stochastiques avec temps local. Probab. Math. Stat. 19(1), 55–62 (1999)

    MathSciNet  MATH  Google Scholar 

  16. Peng, S., Zhu, X.: Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations. Stochastic Process. Appl. 116, 370–380 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Protter, P.: Stochastic integration and differential equations, 2nd edn. Springer, Berlin Heidelberg New York (2003)

    Google Scholar 

  18. Rutkowski, M.: Stochastic differential with singular drift. Stat. Probabil. Lett. 10(3), 225–229 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  19. Samuelson, P.A.: Rational theory of warrant pricing. Ind. Management Rev. 6, 1339 (1965)

    Google Scholar 

  20. Skorokhod, A.V.: Studies in the theory of random processes. Addison-Wesley Publishing Co., Inc, Reading, Mass (1965)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Ibn Tofail University, B.P. 133, Kenitra, Morocco

    Mohsine Benabdallah

  2. LIBMA Laboratory, Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, P.B.O. 2390, Marrakesh, Morocco

    Siham Bouhadou & Youssef Ouknine

Authors
  1. Mohsine Benabdallah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Siham Bouhadou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Youssef Ouknine
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Youssef Ouknine .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Université Cadi Ayyad Department of Mathematics, Marrakesh, Morocco

    M'hamed Eddahbi

  2. Faculté Polydisciplinaire, Université Cadi Ayyad, Safi, Morocco

    El Hassan Essaky

  3. Faculty of Mathematics, University of Barcelona, Barcelona, Spain

    Josep Vives

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Benabdallah, M., Bouhadou, S., Ouknine, Y. (2016). On the Pathwise Uniqueness of Solutions of One-Dimensional Stochastic Differential Equations with Jumps. In: Eddahbi, M., Essaky, E., Vives, J. (eds) Statistical Methods and Applications in Insurance and Finance. CIMPA School 2013. Springer Proceedings in Mathematics & Statistics, vol 158. Springer, Cham. https://doi.org/10.1007/978-3-319-30417-5_8

Download citation

Publish with us

Policies and ethics

Search

Navigation