We present a unified approach to Doob's $L^p$ maximal inequalities for $1\leq
p<\infty$. The novelty of our method is that these martingale inequalities are
obtained as consequences of elementary \emph{deterministic} counterparts. The
latter have a natural interpretation in terms of robust hedging. Moreover our
deterministic inequalities lead to new versions of Doob's maximal inequalities.
These are best possible in the sense that equality is attained by properly
chosen martingales.

We consider the maximization of the long-term growth rate in the
Black-Scholes model under proportional transaction costs as in Taksar, Klass
and Assaf [Math. Oper. Res. 13, 1988]. Similarly as in Kallsen and Muhle-Karbe
[Ann. Appl. Probab., 20, 2010] for optimal consumption over an infinite
horizon, we tackle this problem by determining a shadow price, which is the
solution of the dual problem. It can be calculated explicitly up to determining
the root of a deterministic function.

We give an elementary proof of the celebrated Bichteler-Dellacherie Theorem
which states that the class of stochastic processes $S$ allowing for a useful
integration theory consists precisely of those processes which can be written
in the form $S=M+A$, where $M$ is a local martingale and $A$ is a finite
variation process. In other words, $S$ is a good integrator if and only if it
is a semi-martingale. We obtain this decomposition rather directly from an
elementary discrete-time Doob-Meyer decomposition.

The duality theory of the Monge--Kantorovich transport problem is analyzed in
a general setting. The spaces $X, Y$ are assumed to be polish and equipped with
Borel probability measures $\mu$ and $\nu$. The transport cost function
$c:X\times Y \to [0,\infty]$ is assumed to be Borel. Our main result states
that in this setting there is no duality gap, provided the optimal transport
problem is formulated in a suitably relaxed way.