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