Advertisement

Methodology and Computing in Applied Probability

, Volume 18, Issue 3, pp 691–715 | Cite as

Stochastic Integral Representations of the Extrema of Time-homogeneous Diffusion Processes

  • Runhuan FengEmail author
Article
  • 111 Downloads

Abstract

The stochastic integral representations (martingale representations) of square integrable processes are well-studied problems in applied probability with broad applications in finance. Yet finding explicit expression is not easy and typically done through the Clack-Ocone formula with the advanced machinery of Malliavin calculus. To find an alternative, Shiryaev and Yor (Teor Veroyatnost i Primenen 48(2):375–385, 2003) introduced a relatively simple method using Itô’s formula to develop representations for extrema of Brownian motion. In this paper, we extend their work to provide representations of functionals of time-homogeneous diffusion processes based on the Itô’s formula.

Keywords

Stochastic integral representation Martingale representation Itô formula Time-homogeneous diffusion processes Running extremum 

Mathematics Subject Classification (2010)

Primary 60J60 Secondary 60G17 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Compliance with Ethical Standards

The authors declare that they have no conflict of interest. This research does not involve human participants or animals.

References

  1. Broadie M, Detemple JB (2004) Option pricing: valuation models and applications. Manag Sci 50(9):1145–1177CrossRefGoogle Scholar
  2. Cai N, Li C, Shi C (2014) Closed-form expansions of discretely monitored Asian options in diffusion models. Math Oper Res 39(3):789–822MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chen A, Feng L, Song R (2011) On the monitoring error of the supremum of a normal jump diffusion process. J Appl Probab 48(4):1021–1034MathSciNetCrossRefzbMATHGoogle Scholar
  4. Colwell DB, Elliott RJ, Ekkehard Kopp P (1991) Martingale representation and hedging policies. Stoch Process Appl 38(2):335–345MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cox A, Hobson D, Obloj J (2011) Time-homogeneous diffusions with a given marginal at a random time. ESAIM Probab Stat 15:S11—S24MathSciNetCrossRefGoogle Scholar
  6. Graversen SE, Shiryaev AN (2000) An extension of p. lévy’s distributional properties to the case of a brownian motion with drift. Bernoulli 6(4):615–620MathSciNetCrossRefzbMATHGoogle Scholar
  7. Graversen SE, Shiryaev AN, Yor M (2006) On stochastic integral representations of functionals of Brownian motion. II. Teor Veroyatn Primen 51(1):64–77MathSciNetCrossRefzbMATHGoogle Scholar
  8. Jacobsen M (1996) Laplace and the origin of the ornstein-uhlenbeck process. Bernoulli 2(3):271–286MathSciNetCrossRefzbMATHGoogle Scholar
  9. Janseen A, van Leeuwaarden J (2009) Equidistant sampling for the maximum of a brownian motion with drift on a finite horizon. Electron Commun Probab 14:143–150MathSciNetCrossRefzbMATHGoogle Scholar
  10. Karatzas I, Shreve SE (1991) Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics, 2nd edn. Springer, New YorkGoogle Scholar
  11. Karlin S, Taylor HM (1981) A second course in stochastic processes. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New YorkzbMATHGoogle Scholar
  12. Kimura M (1964) Diffusion models in population genetics. J Appl Probab 1 (2):177–232MathSciNetCrossRefzbMATHGoogle Scholar
  13. Li B, Tang Q, Zhou X (2013) A time-homogenous diffusion model with tax. J Appl Probab 50:195–207MathSciNetCrossRefzbMATHGoogle Scholar
  14. Li B, Zhou X (2013) The joint laplace transforms for diffusion occupation times. Adv Appl Probab 4(895–1201)Google Scholar
  15. Lyulko YA (2010) Stochastic representations of max-type functionals of a random walk. Theory Probab Appl 54(3):516–525MathSciNetCrossRefzbMATHGoogle Scholar
  16. Olver FWJ, Lozier DW, Boisvert RF, Clark CW (eds) (2010) NIST handbook of mathematical functions. U.S. Department of Commerce National Institute of Standards and Technology, Washington, DCGoogle Scholar
  17. Pitman J, Yor M (1999) Laplace transforms related to excursions of a one-dimensioal diffusion. Bernoulli 5(2):249–255MathSciNetCrossRefzbMATHGoogle Scholar
  18. Rémillard B, Renaud J-F (2011) A martingale representation for the maximum of a Lévy process. Commun Stoch Anal 5(4):683–688MathSciNetzbMATHGoogle Scholar
  19. Renaud J-F, Rémillard B (2007) Explicit martingale representations for Brownian functionals and applications to option hedging. Stoch Anal Appl 25(4):801–820MathSciNetCrossRefzbMATHGoogle Scholar
  20. Revuz D, Yor M (1999) Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edn. Springer, BerlinGoogle Scholar
  21. Rogers LCG, Williams D (2000) Diffusions, Markov processes, and martingales. Cambridge mathematical library, vol 2. Cambridge University Press, Cambridge. Itô calculus, Reprint of the second(1994) editionGoogle Scholar
  22. Shiryaev AN, Yor M (2003) On stochastic integral representations of functionals of Brownian motion. I. Teor Veroyatnost i Primenen 48(2):375–385MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.University of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations