Mixing conditions for multivariate infinitely divisible processes with an application to mixed moving averages and the supOU stochastic volatility model

¹ TUM Institute for Advanced Study & Zentrum Mathematik, Technische Universität München, Boltzmannstraße 3, 85748 Garching, Germany
ffuchs@ma.tum.de; www-m4.ma.tum.de
² Institute of Mathematical Finance, Ulm University, Helmholtzstraße 18, 89081 Ulm, Germany
robert.stelzer@uni-ulm.de; www.uni-ulm.de/mawi/finmath.html

Received: 5 May 2011

Abstract

We consider strictly stationary infinitely divisible processes and first extend the mixing conditions given in Maruyama [Theory Probab. Appl. 15 (1970) 1–22] and Rosiński and Żak [Stoc. Proc. Appl. 61 (1996) 277–288] from the univariate to the d-dimensional case. Thereafter, we show that multivariate Lévy-driven mixed moving average processes satisfy these conditions and hence a wide range of well-known processes such as superpositions of Ornstein − Uhlenbeck (supOU) processes or (fractionally integrated) continuous time autoregressive moving average (CARMA) processes are always mixing. Finally, mixing of the log-returns and the integrated volatility process of a multivariate supOU type stochastic volatility model, recently introduced in Barndorff − Nielsen and Stelzer [Math. Finance 23 (2013) 275–296], is established.