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


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.


DIRECT Global optimization Kernel Lévy process Volatility 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Andersen, T.G., Lund, J.: Estimating continuous-time stochastic volatility models of the short-term interest rate. J. Econom. 77, 343–377 (1997) zbMATHCrossRefGoogle Scholar
  2. Barndorff-Nielsen, O.E., Shephard, N.: Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics (with discussion). J. R. Stat. Soc. Ser. B 63, 167–241 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  3. Boole, G.: A Treatise on Differential Equations. Macmillan and Co., London (1872) Google Scholar
  4. Brockwell, P.J.: Heavy-tailed and non-linear continuous-time ARMA models for financial time series. In: Chan, W.S., Li, W.K., Tong, H. (eds.) Statistics and Finance: An Interface, pp. 3–22. Imperial College Press, London (2000) CrossRefGoogle Scholar
  5. Brockwell, P.J.: Lévy-driven CARMA processes. Ann. Inst. Stat. Math. 53, 113–124 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  6. Brockwell, P.J.: Representations of continuous-time ARMA processes. J. Appl. Probab. 41A, 375–382 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  7. Brockwell, P.J., Chadraa, E., Lindner, A.: Continuous time GARCH processes. Ann. Appl. Probab. 16, 790–826 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  8. Brockwell, P.J., Marquardt, T.: Lévy-driven and fractionally integrated ARMA processes with continuous time parameter. Stat. Sin. 15, 477–494 (2005) zbMATHMathSciNetGoogle Scholar
  9. Comte, F., Renault, E.: Long memory in continuous-time stochastic volatility models. Math. Finance 8, 291–323 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  10. Finkel, D.E., Kelley, C.T.: Convergence analysis of the DIRECT algorithm. Technical Report CRSC-TR04-28, Center for Research in Scientific Computation, North Carolina State University (2004). Available at
  11. Gablonsky, J.M.: DIRECT version 2.0. User Guide. Technical Report CRSC-TR01-08, Center for Research in Scientific Computation, North Carolina University (2001) Google Scholar
  12. Gablonsky, J.M., Kelley, C.T.: A locally-biased form of the DIRECT algorithm. J. Glob. Optim. 21, 27–37 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  13. Jones, D.R., Perttunen, C.D., Stuckmann, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79, 157–181 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  14. Kelley, C.T.: Iterative Methods for Optimization. SIAM, Philadelphia (1999) zbMATHGoogle Scholar
  15. Klüppelberg, C., Lindner, A., Maller, R.: A continuous time GARCH process driven by a Lévy process: stationarity and second order behaviour. J. Appl. Probab. 41, 601–622 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  16. Roberts, G.O., Papaspiliopoulos, O., Dellaportas, P.: Bayesian inference for non-Gaussian Ornstein-Uhlenbeck stochastic volatility processes. J. R. Stat. Soc. Ser. B 66, 369–393 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  17. Todorov, V., Tauchen, G.: Simulation methods for Lévy-driven CARMA stochastic volatility models. J. Bus. Econ. Statistics 24, 455–469 (2006) CrossRefMathSciNetGoogle Scholar
  18. Tsai, H., Chan, K.S.: A note on non-negative continuous-time processes. J. R. Stat. Soc. Ser. B 67, 589–597 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  19. Tsai, H., Chan, K.S.: A Note on the Non-negativity of Continuous-time ARMA and GARCH Processes. Technical Report No. 359, Department of Statistics and Actuarial Science, The University of Iowa (2006). Downloadable from

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

Personalised recommendations