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October 2008 Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping
A. M. G. Cox, David Hobson, Jan Obłój
Ann. Appl. Probab. 18(5): 1870-1896 (October 2008). DOI: 10.1214/07-AAP507


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 $\sup_{\tau}\mathbb{E}[F(L_{\tau})-\int _{0}^{\tau}\beta(B_{s})\,ds]$. 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.


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A. M. G. Cox. David Hobson. Jan Obłój. "Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping." Ann. Appl. Probab. 18 (5) 1870 - 1896, October 2008.


Published: October 2008
First available in Project Euclid: 30 October 2008

zbMATH: 1165.60020
MathSciNet: MR2462552
Digital Object Identifier: 10.1214/07-AAP507

Primary: 60G40
Secondary: 60G44, 91B28

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.18 • No. 5 • October 2008
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