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

Abstract

We develop a class of pathwise inequalities of the form H(B_t)≥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.