A Note on the Non-negativity of Continuous-time ARMA and GARCH Processes

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 1 of Continuous-time Auto-Regressive Moving-Average (CARMA) processes, we derive a sufficient condition for the kernel to be non-negative, based on which we propose a numerical method for checking the non-negativity of a kernel function. We discuss how to adapt this approach to solving a similar problem with the second approach to modeling volatility via the COntinuoustime Generalized Auto-Regressive Conditional Heteroscedastic (COGARCH) processes