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We are interested in proposing approximations of a sequence of probabilitymeasures in the convex order by finitely supported probability measures stillin the convex order. We propose to alternate transitions according to amartingale Markov kernel mapping a probability measure in the sequence to thenext and dual quantization steps. In the case of ARCH models and in particularof the Euler scheme of a driftless Brownian diffusion, the noise has to betruncated to enable the dual quantization step. We analyze the error betweenthe original ARCH model and its approximation with truncated noise and exhibitconditions under which the latter is dominated by the former in the convexorder at the level of sample-paths. Last, we analyse the error of the schemecombining the dual quantization steps with truncation of the noise according toprimal quantization.

Convex order, quantization and monotone approximations of ARCH models