A note on the non-negativity of continuous-time ARMA and GARCH processes

Authors:
Henghsiu Tsai;Kung-Sik Chan
Affiliations:
Institute of Statistical Science, Academia Sinica, Taipei, Taiwan 115;Department of Statistics and Actuarial Science, University of Iowa, Iowa City, USA 52242
Venue:
Statistics and Computing
Year:
2009

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Abstract

A general approach for modeling the volatility process in continuous-time is based on the convolution of a kernel with a non-decreasing Lévy process, which is non-negative if the kernel is non-negative. Within the framework of Continuous-time Auto-Regressive Moving-Average (CARMA) processes, we derive a necessary condition for the kernel to be non-negative, and propose a numerical method for checking the non-negativity of a kernel function. These results can be lifted to solving a similar problem with another approach to modeling volatility via the COntinuous-time Generalized Auto-Regressive Conditional Heteroscedastic (COGARCH) processes.