Skip to main content
SpringerLink
Log in

A Discrete-Time Clark–Ocone Formula and its Application to an Error Analysis

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

In this paper, we will establish a discrete-time version of Clark(–Ocone–Haussmann) formula, which can be seen as an asymptotic expansion in a weak sense. The formula is applied to the estimation of the error caused by the martingale representation. Throughout, we use another distribution theory with respect to Gaussian rather than Lebesgue measure, which can be seen as a discrete Malliavin calculus.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bertsimas, D., Kogan, L., Lo, A.W.: When is time continuous? J. Financ. Econ. 55, 173–204 (2000)

    Article  MATH  Google Scholar 

  2. Billingsley, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics, Second edn. A Wiley-Interscience Publication. Wiley, New York (1999)

    Book  MATH  Google Scholar 

  3. Clark, J.M.C.: The representation of functionals of Brownian motion by stochastic integral. Ann. Math. Stat. 41, 1282–1295 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  4. Geiss, C., Steinicke, A.: \(L_{2}\)-variation of Lévy driven BSDEs with non-smooth terminal conditions. Bernoulli 22(2), 995–1025 (2016)

  5. Geiss, C., Geiss, S., Gobet, E.: Generalized fractional smoothness and Lp-variation of BSDEs with non-Lipschitz terminal condition. Stoch. Process. Appl. 122(5), 2078–2116 (2012)

    Article  MATH  Google Scholar 

  6. Geiss, S., Geiss, C.: On approximation of a class of stochastic integrals and interpolation. Stoch. Stoch. Rep. 76, 339–362 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Geiss, S., Hujo, M.: Interpolation and approximation in \(L_{2}(\gamma )\). J. Approx. Theory 144(2), 213–232 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Geiss, S., Toivola, A.: Weak convergence of error processes in discretizations of stochastic integrals and Besov spaces. Bernoulli 15(4), 925–954 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gobet, E., Temam, E.: Discrete time hedging errors for options with irregular pay-offs. Finance Stoch. 5(3), 357–367 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Haussmann, U.G.: On the integral representation of functionals of Itô processes. Stochastics 3(1), 17–27 (1979)

    MathSciNet  MATH  Google Scholar 

  11. Hayashi, T., Mykland, P.A.: Evaluating hedging errors: an asymptotic approach. Math. Finance 15(2), 309–343 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland (1981)

  13. Malliavin, P., Thalmaier, A.: Stochastic Calculus of Variations in Mathematical Finance. Springer Finance. Springer, Berlin (2006)

    MATH  Google Scholar 

  14. Ngo, H.L., Ogawa, S.: On the discrete approximation of occupation time of diffusion processes. Electron. J. Stat. 5, 1374–1393 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nualart, D., Peccati, G.: Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33(1), 177–193 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ocone, D.: Malliavin’s calculus and stochastic integral representations of functionals of diffusion processes. Stochastics 12(3–4), 161–185 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Privault, N.: Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales. Lecture Notes in Mathematics. Springer, Berlin (1982)

    Google Scholar 

  18. Renaud, J.F., Rémillard, B.: Explicit martingale representations for Brownian functionals and applications to option hedging. Stoch. Anal. Appl. 25, 810–820 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rootzen, H.: Limit distributions for the error in approximations of stochastic integrals. Ann. Probab. 8(2), 241–251 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)

    MATH  Google Scholar 

  21. Temam, E.: Analysis of error with Malliavin calculus: application to hedging. Math. Finance 13(1), 201–214 (2003)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan

    Jirô Akahori, Takafumi Amaba & Kaori Okuma

Authors
  1. Jirô Akahori
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takafumi Amaba
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kaori Okuma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Takafumi Amaba.

Additional information

Jirô Akahori: The First author was supported by JSPS KAKENHI Grant Number 23330109, 24340022, 23654056, 25285102 and the project RARE-318984 (an FP7 Marie Curie IRSES).

Takafumi Amaba: This work was supported by JSPS KAKENHI Grant Number 24\(\cdot \)5772.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Akahori, J., Amaba, T. & Okuma, K. A Discrete-Time Clark–Ocone Formula and its Application to an Error Analysis. J Theor Probab 30, 932–960 (2017). https://doi.org/10.1007/s10959-016-0666-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-016-0666-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation