We investigate the (functional) convex order of for various continuous martingale processes, either with respect to their diffusions coefficients for Lévy-driven SDEs or their integrands for stochastic integrals. Main results are bordered by counterexamples. Various upper and lower bounds can be derived for path wise European option prices in local volatility models. In view of numerical applications, we adopt a systematic (and symmetric) methodology: (a) propagate the convexity in a {\em simulatable} dominating/dominated discrete time model through a backward induction (or linear dynamical principle); (b) Apply functional weak convergence results to numerical schemes/time discretizations of the continuous time martingale satisfying (a) in order to transfer the convex order properties. Various bounds are derived for European options written on convex pathwise dependent payoffs. We retrieve and extend former results obtains by several authors since the seminal 1985 paper by Hajek . In a second part, we extend this approach to Optimal Stopping problems using a that the Snell envelope satisfies (a') a Backward Dynamical Programming Principle to propagate convexity in discrete time; (b') satisfies abstract convergence results under non-degeneracy assumption on filtrations. Applications to the comparison of American option prices on convex pathwise payoff processes are given obtained by a purely probabilistic arguments.