Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 21 (2026), 107 -- 156
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

THE FUNDAMENTALS OF GARCH

Jorge Bustamante and Adrian Pizzinga

Abstract. This paper goes over very first theoretical aspects regarding the GARCH process. Our survey comprises: (i) a literature review that indicates the parts of the founding theory covered by many standard textbooks (and some other parts that are not); and (ii) detailed and classroom level proofs of the existence and uniqueness of a stationary solution for the GARCH defining equations. We aim to contribute to the syllabus of typical graduate level courses on time series analysis, time series econometrics, financial econometrics, statistical methods in finance, risk theory, etc. The target audience includes both students and lecturers from these courses.

2020 Mathematics Subject Classification: 60G10; 60G12; 97K50.
Keywords: GARCH process; stationarity; existence; P-almost surely uniqueness.

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Jorge Bustamante, ORCID: 0009-0007-7916-9712
Department of Economics
Pontifical Catholic University of Rio de Janeiro, Brazil.
e-mail: jroqueb95@gmail.com



Adrian Pizzinga - Corresponding author, ORCID: 0000-0002-3059-743X
Institute of Mathematics and Statistics
Fluminense Federal University, Brazil.
e-mail: adrianhp@id.uff.br






Received: December 27, 2025; Accepted: April 16, 2026;
Published electronically: April 17, 2026.