  In this paper we introduce and study the concept of optimal and surely
optimal dual martingales in the context of dual valuation of Bermudan options,
and outline the development of new algorithms in this context. We provide a
characterization theorem, a theorem which gives conditions for a martingale to
be surely optimal, and a stability theorem concerning martingales which are
near to be surely optimal in a sense. Guided by these results we develop a
framework of backward algorithms for constructing such a martingale. In turn
this martingale may then be utilized for computing an upper bound of the
Bermudan product. The methodology is pure dual in the sense that it doesn't
require certain input approximations to the Snell envelope. In an It\^o-L\'evy
environment we outline a particular regression based backward algorithm which
allows for computing dual upper bounds without nested Monte Carlo simulation.
Moreover, as a by-product this algorithm also provides approximations to the
continuation values of the product, which in turn determine a stopping policy.
Hence, we may obtain lower bounds at the same time. In a first numerical study
we demonstrate the backward dual regression algorithm in a Wiener environment
at well known benchmark examples. It turns out that the method is at least
comparable to the one in Belomestny et. al. (2009) regarding accuracy, but
regarding computational robustness there are even several advantages.
