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Adaptive nonparametric drift estimation for multivariate jump diffusions under sup-norm risk

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  • Dexheimer, Niklas

Abstract

We investigate nonparametric drift estimation for multidimensional jump diffusions based on continuous observations. The results are derived under anisotropic smoothness assumptions and the estimators’ performance is measured in terms of the sup-norm loss. We present two different Nadaraya–Watson type estimators, which are both shown to achieve the minimax optimal classical nonparametric rate of convergence under varying assumptions on the jump measure. Fully data-driven versions of both estimators are also introduced and shown to attain the same rate of convergence. The results rely on novel uniform moment bounds for empirical processes associated to the investigated jump diffusion, which are of independent interest.

Suggested Citation

  • Dexheimer, Niklas, 2025. "Adaptive nonparametric drift estimation for multivariate jump diffusions under sup-norm risk," Stochastic Processes and their Applications, Elsevier, vol. 190(C).
    • Handle: RePEc:eee:spapps:v:190:y:2025:i:c:s030441492500184x
    DOI: 10.1016/j.spa.2025.104741
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    References listed on IDEAS

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    1. Claire Lacour & Pascal Massart & Vincent Rivoirard, 2017. "Estimator Selection: a New Method with Applications to Kernel Density Estimation," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(2), pages 298-335, August.
    2. Schmisser, Émeline, 2014. "Non-parametric adaptive estimation of the drift for a jump diffusion process," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 883-914.
    3. Dexheimer, Niklas & Strauch, Claudia, 2022. "Estimating the characteristics of stochastic damping Hamiltonian systems from continuous observations," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 321-362.
    4. Wang, Hanchao & Lin, Zhengyan & Su, Zhonggen, 2019. "On Bernstein type inequalities for stochastic integrals of multivariate point processes," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1605-1621.
    5. Mancini, Cecilia & Renò, Roberto, 2011. "Threshold estimation of Markov models with jumps and interest rate modeling," Journal of Econometrics, Elsevier, vol. 160(1), pages 77-92, January.
    6. Park, Joon Y. & Wang, Bin, 2021. "Nonparametric estimation of jump diffusion models," Journal of Econometrics, Elsevier, vol. 222(1), pages 688-715.
    7. Comte, Fabienne & Prieur, Clémentine & Samson, Adeline, 2017. "Adaptive estimation for stochastic damping Hamiltonian systems under partial observation," Stochastic Processes and their Applications, Elsevier, vol. 127(11), pages 3689-3718.
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