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

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

2007-02-07

arxiv.org/abs/math/0702173v2

Annals of Applied Probability 2008, Vol. 18, No. 5, 1870-1896

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AAP507 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/07-AAP507

We develop a class of pathwise inequalities of the form $H(B_t)\ge M_t+F(L_t)$, 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 $\sup_{\tau}\mathbb{E}[F(L_{\tau})-\int _0^{\tau}\beta(B_s) ds]$. 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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