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Orthogonal Expansion of Levy Process Functionals: Theory and Practice

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  • Chaohua Dong

  • Jiti Gao

Abstract

In this paper, expansions of functionals of Levy processes are established under some Hilbert spaces and their orthogonal bases. From practical standpoint, both time-homogeneous and time-inhomogeneous functionals of Levy processes are considered. Several expansions and rates of convergence are established. In order to state asymptotic distributions for statistical estimators of unknown parameters involved in a general regression model, we develop a general asymptotic theory for partial sums of functionals of Levy processes. The results show that these estimators of the unknown parameters in different situations converge to quite different random variables. In addition, the rates of convergence depend on various factors rather than just the sample size. Simulations and empirical study are provided to illustrate the theoretical results.

Suggested Citation

  • Chaohua Dong & Jiti Gao, 2013. "Orthogonal Expansion of Levy Process Functionals: Theory and Practice," Monash Econometrics and Business Statistics Working Papers 3/13, Monash University, Department of Econometrics and Business Statistics.
    • Handle: RePEc:msh:ebswps:2013-3
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    File URL: http://business.monash.edu/econometrics-and-business-statistics/research/publications/ebs/wp03-13.pdf
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    References listed on IDEAS

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    1. Jiti Gao & Peter C.B. Phillips, 2011. "Semiparametric Estimation in Multivariate Nonstationary Time Series Models," Monash Econometrics and Business Statistics Working Papers 17/11, Monash University, Department of Econometrics and Business Statistics.
    2. Gao, Jiti, 2007. "Nonlinear time series: semiparametric and nonparametric methods," MPRA Paper 39563, University Library of Munich, Germany, revised 01 Sep 2007.
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    4. Dong, Chaohua & Gao, Jiti, 2013. "Solving replication problems in a complete market by orthogonal series expansion," The North American Journal of Economics and Finance, Elsevier, vol. 25(C), pages 306-317.
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    7. Qi Li & Jeffrey Scott Racine, 2006. "Nonparametric Econometrics: Theory and Practice," Economics Books, Princeton University Press, edition 1, volume 1, number 8355.
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