In this paper, we present an approximation scheme to solve optimal stopping problems based on fully non-Markovian reward continuous processes adapted to the filtration generated by the multi-dimensional Brownian motion. The approximations satisfy suitable variational inequalities which allow us to construct $\epsilon$-optimal stopping times and optimal values in full generality. More importantly, the methodology allows us to design concrete Monte-Carlo schemes for non-Markovian optimal stopping time problems as demonstrated in the companion paper [5] . The framework is applied to path-dependent SDEs driven by Brownian motion and to SDEs with additive noise driven by fractional Brownian motion.
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