Convex order, quantization and monotone approximations of ARCH models

TL;DR

This paper introduces a novel method for approximating sequences of probability measures in the convex order using dual quantization and martingale kernels, with applications to ARCH models and Brownian diffusions, analyzing errors and dominance conditions.

Contribution

It proposes a new approximation scheme for convex order probability measures combining dual quantization, martingale kernels, and noise truncation, specifically applied to ARCH models.

Findings

Error bounds between original and approximated models are established.

Conditions for convex order dominance in sample paths are identified.

The scheme effectively combines dual and primal quantization with noise truncation.

Abstract

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 transitions according to a martingale Markov kernel mapping a probability measure in the sequence to the next and dual quantization steps. In the case of 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 analyse the error of the scheme combining the dual quantization steps with truncation of the noise according to primal quantization.