Conditions for the existence of strictly stationary multivariate GARCH processes in the so-called BEKK parametrisation, which is the most general form of multivariate GARCH processes typically used in applications, and for their geometric ergodicity are obtained. The conditions are that the driving noise is absolutely continuous with respect to the Lebesgue measure and zero is in the interior of its support and that a certain matrix built from the GARCH coefficients has spectral radius smaller than one. To establish the results, semi-polynomial Markov chains are defined and analysed using algebraic geometry. © 2011 Elsevier B.V. All rights reserved.
Boussama, F., Fuchs, F., & Stelzer, R. (2011). Stationarity and geometric ergodicity of BEKK multivariate GARCH models. Stochastic Processes and Their Applications, 121(10), 2331–2360. https://doi.org/10.1016/j.spa.2011.06.001