Convex Order, Quantization and Monotone Approximations of ARCH Models

Benjamin Jourdain () and Gilles Pagès ()
Additional contact information
Benjamin Jourdain: Cermics, Ecole des Ponts, INRIA
Gilles Pagès: UMR 8001, Campus Pierre et Marie Curie, Sorbonne Université case 158

Journal of Theoretical Probability, 2022, vol. 35, issue 4, 2480-2517

Abstract: 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 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.

Keywords: Quantization; Convex order; Martingales; ARCH models; 60G42; 94A29; 62M10 (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

Downloads: (external link)
http://link.springer.com/10.1007/s10959-021-01141-1 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:35:y:2022:i:4:d:10.1007_s10959-021-01141-1

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-021-01141-1

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().