We propose two signature-based methods to solve an 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 generalisation of the famous Longstaff–Schwartz algorithm, using linear functionals of the signature as regression basis. For the dual formulation, we parametrise 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.
