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Statistics and Computing

, Volume 19, Issue 2, pp 149–153 | Cite as

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

  • Henghsiu TsaiEmail author
  • Kung-Sik Chan
Article

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.

Keywords

DIRECT Global optimization Kernel Lévy process Volatility 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institute of Statistical ScienceAcademia SinicaTaipeiTaiwan
  2. 2.Department of Statistics and Actuarial ScienceUniversity of IowaIowa CityUSA

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