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

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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.

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Authors and Affiliations

  1. Institute of Statistical Science, Academia Sinica, Taipei, 115, Taiwan

    Henghsiu Tsai

  2. Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA, 52242, USA

    Kung-Sik Chan

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  1. Henghsiu Tsai
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  2. Kung-Sik Chan
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Correspondence to Henghsiu Tsai.

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Tsai, H., Chan, KS. A note on the non-negativity of continuous-time ARMA and GARCH processes. Stat Comput 19, 149–153 (2009). https://doi.org/10.1007/s11222-008-9078-7

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  • DOI: https://doi.org/10.1007/s11222-008-9078-7

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