Convex Order, Quantization and Monotone Approximations of ARCH Models

We are interested in proposing approximations of a sequence of probability measures in the convex order by finitely supported probability measures still in the convex order. We propose to alternate two types of operators: transition according to a one-step martingale Markov kernel mapping a probability measure in the sequence to its successor and spatial discretization through dual (also called Delaunay) quantization. In the case of autoregressive conditional heteroskedasticity (ARCH) models and in particular of the Euler scheme of a driftless Brownian diffusion, the noise has to be truncated to enable the dual quantization step. We analyze the error between the original ARCH model and its approximation with truncated noise and exhibit conditions under which the latter is dominated by the former in the convex order at the level of sample paths. Last, we analyze the error of the scheme combining the dual quantization steps with truncation of the noise according to primal quantization.