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


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.


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

Mathematics Subject Classification (2010)

Primary 60J60 Secondary 60G17 


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Compliance with Ethical Standards

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


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.University of Illinois at Urbana-ChampaignUrbanaUSA

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