  We investigate the (functional) convex order of for various continuous
martingale processes, either with respect to their diffusions coefficients for
L\'evy-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.
