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Pathwise inequalities for local time : applications to skorokhod embeddings and optimal stopping

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Cox, A. M. G., Hobson, David (David G.) and Obłój, Jan. (2008) Pathwise inequalities for local time : applications to skorokhod embeddings and optimal stopping. Annals of Applied Probability, Vol.18 (No.5). pp. 1870-1896. ISSN 1050-5164

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Official URL: http://dx.doi.org/10.1214/07-AAP507

Abstract

We develop a class of pathwise inequalities of the form H(B-t) >= M-t + F(L-t), where B-t is Brownian motion, L-t its local time at zero and M-t a local martingale. The concrete nature of the representation makes the inequality useful for a variety of applications. In this work, we use the inequalities to derive constructions and optimality results of Vallois' Skorokhod embeddings. We discuss their financial interpretation in the context of robust pricing and hedging of options written on the local time. In the final part of the paper we use the inequalities to solve a class of optimal stopping problems of the form sup(tau) E[F(L-tau) - integral(t)(0) beta(B-s)ds]. The solution is given via a minimal solution to a system of differential equations and thus resembles the maximality principle described by Peskir. Throughout, the emphasis is placed on the novelty and simplicity of the techniques.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Local times (Stochastic processes), Brownian motion processes, Martingales (Mathematics), Optimal stopping (Mathematical statistics)
Journal or Publication Title:	Annals of Applied Probability
Publisher:	Institute of Mathematical Statistics
ISSN:	1050-5164
Date:	October 2008
Volume:	Vol.18
Number:	No.5
Number of Pages:	27
Page Range:	pp. 1870-1896
Identification Number:	10.1214/07-AAP507
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access
Funder:	Nuffield Foundation (NF), Engineering and Physical Sciences Research Council (EPSRC), Sixth Framework Programme (European Commission) (FP6)
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URI:	http://wrap.warwick.ac.uk/id/eprint/29107