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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. 18701896. ISSN 10505164

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Official URL: http://dx.doi.org/10.1214/07AAP507
Abstract
We develop a class of pathwise inequalities of the form H(Bt) >= Mt + F(Lt), where Bt is Brownian motion, Lt its local time at zero and Mt 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(Ltau)  integral(t)(0) beta(Bs)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:  10505164  
Official Date:  October 2008  
Dates: 


Volume:  Vol.18  
Number:  No.5  
Number of Pages:  27  
Page Range:  pp. 18701896  
Identification Number:  10.1214/07AAP507  
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)  
References:  [1] ALILI, L. and KYPRIANOU, A. E. (2005). Some remarks on first passage of Lévy processes, 

URI:  http://wrap.warwick.ac.uk/id/eprint/29107 
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