The Annals of Applied Probability

Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping

A. M. G. Cox, David Hobson, and Jan Obłój

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

Article information

Source
Ann. Appl. Probab. Volume 18, Number 5 (2008), 1870-1896.

Dates
First available in Project Euclid: 30 October 2008

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1225372954

Digital Object Identifier
doi:10.1214/07-AAP507

Zentralblatt MATH identifier
05374757

Mathematical Reviews number (MathSciNet)
MR2462552

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60G44: Martingales with continuous parameter 91B28

Keywords
Skorokhod embedding problem local time Vallois stopping time optimal stopping robust pricing and hedging

Citation

Cox, A. M. G.; Hobson, David; Obłój, Jan. Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping. The Annals of Applied Probability 18 (2008), no. 5, 1870--1896. doi:10.1214/07-AAP507. http://projecteuclid.org/euclid.aoap/1225372954.


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