2007/02/28 by A. M. G. Cox, David Hobson, Jan Obłój
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #Capital Investment and Risk Analysis #Probability and Risk Models #Stochastic processes and financial applications #math.PR #msc:60G40 #msc:60G44 #msc:91B28 #q-fin.PR
paper · pdf · doi:10.1214/07-aap507
published as Annals of Applied Probability 2008, Vol. 18, No. 5, 1870-1896 · Published in at http://dx.doi.org/10.1214/07-AAP507 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2008/10/01 · arxiv created 2008/11/13 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 supτ𝔼[F(Lτ)-∫ 0τβ(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.