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On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions

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  • Löcherbach, Eva
  • Loukianova, Dasha

Abstract

We introduce a sequence of stopping times that allow us to study an analogue of a life-cycle decomposition for a continuous time Markov process, which is an extension of the well-known splitting technique of Nummelin to the continuous time case. As a consequence, we are able to give deterministic equivalents of additive functionals of the process and to state a generalisation of Chen's inequality. We apply our results to the problem of non-parametric kernel estimation of the drift of multi-dimensional recurrent, but not necessarily ergodic, diffusion processes.

Suggested Citation

  • Löcherbach, Eva & Loukianova, Dasha, 2008. "On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 118(8), pages 1301-1321, August.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:8:p:1301-1321
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    References listed on IDEAS

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    1. Khasminskii, R., 2001. "Limit distributions of some integral functionals for null-recurrent diffusions," Stochastic Processes and their Applications, Elsevier, vol. 92(1), pages 1-9, March.
    2. Yu. Kutoyants, 1998. "Efficient Density Estimation for Ergodic Diffusion Processes," Statistical Inference for Stochastic Processes, Springer, vol. 1(2), pages 131-155, May.
    3. R. Höpfner & Yu. Kutoyants, 2003. "On a Problem of Statistical Inference in Null Recurrent Diffusions," Statistical Inference for Stochastic Processes, Springer, vol. 6(1), pages 25-42, January.
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    Cited by:

    1. Kanaya, Shin, 2017. "Uniform Convergence Rates Of Kernel-Based Nonparametric Estimators For Continuous Time Diffusion Processes: A Damping Function Approach," Econometric Theory, Cambridge University Press, vol. 33(4), pages 874-914, August.
    2. Löcherbach, Eva & Loukianova, Dasha, 2009. "The law of iterated logarithm for additive functionals and martingale additive functionals of Harris recurrent Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2312-2335, July.
    3. Eva Löcherbach & Dasha Loukianova, 2012. "Deviation Inequalities for Centered Additive Functionals of Recurrent Harris Processes Having General State Space," Journal of Theoretical Probability, Springer, vol. 25(1), pages 231-261, March.
    4. Rey, Clément, 2019. "Approximation of Markov semigroups in total variation distance under an irregular setting: An application to the CIR process," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 539-571.

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