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Nonparametric estimation of the transition density function for diffusion processes

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  • Comte, Fabienne
  • Marie, Nicolas

Abstract

We assume that we observe N∈N∗ independent copies of a diffusion process on a time-interval [0,2T]. For a given time t∈(0,T], we estimate the transition density pt(x,.), namely the conditional density of Xt+s given Xs=x, under conditions on the diffusion coefficients ensuring that this quantity exists. We use a least squares projection method on a product of finite dimensional spaces, prove risk bounds for the estimator and propose an anisotropic model selection method, relying on several reference norms. A simulation study illustrates the theoretical part for Ornstein–Uhlenbeck or square-root (Cox-Ingersoll-Ross) processes.

Suggested Citation

  • Comte, Fabienne & Marie, Nicolas, 2025. "Nonparametric estimation of the transition density function for diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 188(C).
    • Handle: RePEc:eee:spapps:v:188:y:2025:i:c:s0304414925001085
    DOI: 10.1016/j.spa.2025.104667
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    References listed on IDEAS

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    1. Lacour, Claire, 2008. "Nonparametric estimation of the stationary density and the transition density of a Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 232-260, February.
    2. Denis Belomestny & Fabienne Comte & Valentine Genon-Catalot, 2019. "Sobolev-Hermite versus Sobolev nonparametric density estimation on $${\mathbb {R}}$$ R," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(1), pages 29-62, February.
    3. 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.
    4. Nicolas Marie, 2023. "Nonparametric estimation for i.i.d. paths of a martingale-driven model with application to non-autonomous financial models," Finance and Stochastics, Springer, vol. 27(1), pages 97-126, January.
    5. Nicolas Marie, 2025. "On a computable Skorokhod's integral‐based estimator of the drift parameter in fractional SDE," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 52(1), pages 1-37, March.
    6. F. Comte & V. Genon-Catalot, 2020. "Regression function estimation as a partly inverse problem," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(4), pages 1023-1054, August.
    7. Christophe Denis & Charlotte Dion & Miguel Martinez, 2020. "Consistent procedures for multiclass classification of discrete diffusion paths," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(2), pages 516-554, June.
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