Windings of planar processes, exponential functionals and Asian options

Motivated by a common mathematical finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of Brownian motion in the framework of planar Brownian motion. We prove a conjecture of Vakeroudis and Yor (2012) co...

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Bibliographic Details
Published inAdvances in applied probability Vol. 50; no. 3; pp. 726 - 742
Main Authors Jedidi, Wissem, Vakeroudis, Stavros
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.09.2018
Applied Probability Trust
Subjects
Asymptotic properties
Brownian motion
Coils (windings)
Economic models
Original Article
Probability
Random variables
stable process
60J65
60G44
Bessel clock
Bernstein function
infinite divisibility
60G52
Lévy process
Asian option
Bougerol's identity
winding
Planar Brownian motion
Lévy measure
Secondary 60F05
Primary 60G51
skew-product representation
91G80
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Summary:Motivated by a common mathematical finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of Brownian motion in the framework of planar Brownian motion. We prove a conjecture of Vakeroudis and Yor (2012) concerning infinite divisibility properties of this random variable and present a novel simple proof of the result of DeBlassie (1987), (1988) concerning the asymptotic behavior of the distribution of the Bessel clock appearing in the skew-product representation of planar Brownian motion, as t→∞. We use the results of the windings approach in order to obtain results for quantities associated to the pricing of Asian options.
ISSN:0001-8678
1475-6064
DOI:10.1017/apr.2018.33