  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.
