  We propose two signature-based methods to solve the optimal stopping problem
- that is, to price American options - in non-Markovian frameworks. Both
methods rely on a global approximation result for $L^p-$functionals on rough
path-spaces, using linear functionals of robust, rough path signatures. In the
primal formulation, we present a non-Markovian generalization of the famous
Longstaff-Schwartz algorithm, using linear functionals of the signature as
regression basis. For the dual formulation, we parametrize the space of
square-integrable martingales using linear functionals of the signature, and
apply a sample average approximation. We prove convergence for both methods and
present first numerical examples in non-Markovian and non-semimartingale
regimes.
