We aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for d = 1 and d = 2. We consider a class of fully non-linear jump diffusion processes whose invariant density belongs to some Hölder space. Firstly, in dimension one, we show that the kernel density estimator achieves the convergence rate $$, which is the optimal rate in the absence of jumps. This improves the convergence rate obtained in Amorino and Gloter [J. Stat. Plann. Inference 213 (2021) 106–129], which depends on the Blumenthal-Getoor index for d = 1 and is equal to $$ for d = 2. Secondly, when the jump and diffusion coefficients are constant and the jumps are finite, we show that is not possible to find an estimator with faster rates of estimation. Indeed, we get some lower bounds with the same rates $$ in the mono and bi-dimensional cases, respectively. Finally, we obtain the asymptotic normality of the estimator in the one-dimensional case for the fully non-linear process.
Keywords: Minimax risk, convergence rate, non-parametric statistics, ergodic diffusion with jumps, Lévy driven SDE, invariant density estimation
@article{PS_2022__26_1_126_0,
author = {Amorino, Chiara and Nualart, Eulalia},
title = {Optimal convergence rates for the invariant density estimation of jump-diffusion processes},
journal = {ESAIM: Probability and Statistics},
pages = {126--151},
year = {2022},
publisher = {EDP-Sciences},
volume = {26},
doi = {10.1051/ps/2022001},
mrnumber = {4373279},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2022001/}
}
TY - JOUR AU - Amorino, Chiara AU - Nualart, Eulalia TI - Optimal convergence rates for the invariant density estimation of jump-diffusion processes JO - ESAIM: Probability and Statistics PY - 2022 SP - 126 EP - 151 VL - 26 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ps/2022001/ DO - 10.1051/ps/2022001 LA - en ID - PS_2022__26_1_126_0 ER -
%0 Journal Article %A Amorino, Chiara %A Nualart, Eulalia %T Optimal convergence rates for the invariant density estimation of jump-diffusion processes %J ESAIM: Probability and Statistics %D 2022 %P 126-151 %V 26 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ps/2022001/ %R 10.1051/ps/2022001 %G en %F PS_2022__26_1_126_0
Amorino, Chiara; Nualart, Eulalia. Optimal convergence rates for the invariant density estimation of jump-diffusion processes. ESAIM: Probability and Statistics, Tome 26 (2022), pp. 126-151. doi: 10.1051/ps/2022001
[1] and , Invariant density adaptive estimation for ergodic jump diffusion processes over anisotropic classes. J. Stat. Plan. Inference 213 (2021) 106–129. | MR | DOI
[2] , Rate of estimation for the stationary distribution of jump-processes over anisotropic Hölder classes. To appear Electr. J. Stat. (2022). | MR
[3] , , and , On the nonparametric inference of coefficients of self-exciting jump-diffusion. Preprint (2020). | arXiv | MR
[4] , Lévy processes and stochastic calculus. Cambridge University Press (2009). | MR | Zbl | DOI
[5] , Nonparametric identification for diffusion processes. SIAM J. Control Optim. 16 (1978) 380–395. | MR | Zbl | DOI
[6] and , Recursive estimation in diffusion model. SIAM J. Control Optim. 19 (1981) 676–685. | MR | Zbl | DOI
[7] and , Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc. Ser. B, Stat. Methodol. 63 (2001) 167–241. | MR | Zbl | DOI
[8] , Estimation de la densité pour des trajectoires non directement observables. Publ. Inst. Statist. Univ. Paris 40 (1996) 21–36. | MR | Zbl
[9] , Parametric rates of nonparametric estimators and predictors for continuous time processes. Ann. Statist. 25 (1997) 982–1000. | MR | Zbl | DOI
[10] , Nonparametric Statistics for Stochastic Processes (Second edition), Lecture Notes Statist., 110. Springer-Verlag, New York (1998). | MR | Zbl | DOI
[11] , Minimax rates of density estimators for continuous time processes. Sankhya. Ser. A 60 (1998) 18–28. | MR | Zbl
[12] and , Local time and density estimation in continuous time. Math. Methods Stat. 8 (1999) 22–45. | MR | Zbl
[13] , and , Asymptotic normality for density kernel estimators in discrete and continuous time. J. Multivar. Anal. 68 (1999) 78–95. | MR | Zbl | DOI
[14] and , On smoothed probability density estimation for stationary processes. Stoch. Process Appl. 21 (1986) 179–193. | MR | Zbl | DOI
[15] , , and , Heat kernels for non-symmetric diffusion operators with jumps. J Differ Equ. 263 (2017) 6576–6634. | MR | DOI
[16] , Nonparametric regression and prediction for continuous-time processes. Publ. Inst. Statist. Univ. Paris 38 (1994) 37–58. | MR | Zbl
[17] and , Anisotropic adaptive kernel deconvolution. Ann. Inst. Henri Poincaré, Probab. Statis. 49 (2013) 569–609. | MR | Numdam | Zbl
[18] and , Super optimal rates for nonparametric density estimation via projection estimators. Stoch. Process Appl. 115 (2005) 797–826. | MR | Zbl | DOI
[19] , and , Rate of Estimation for the Stationary Distribution of Stochastic Damping Hamiltonian Systems with Continuous Observations. Preprint (2020). | arXiv | MR
[20] , Sur l’estimation des densités d’un processus stationnaire á temps continu. Publications de l’ISUP XXV (1980) 17–39. | MR | Zbl
[21] , Efficient density estimation for ergodic diffusion processes. Stat. Inference Stoch. Process. 1 (1998) 131–155. | MR | Zbl | DOI
[22] , and , Mixing it up: A general framework for Markovian statistics beyond reversibility and the minimax paradigm. Preprint (2020). | arXiv
[23] and , Nonparametric drift estimation for diffusions with jumps driven by a Hawkes process. Stat. Inference Stochastic Processes 23 (2020) 489–515. | MR | DOI
[24] and , The Morris–Lecar neuron model embeds a leaky integrate-and-fire model. J. Math. Biol. 67 (2013) 239–259. | MR | Zbl | DOI
[25] , Mixing: properties and examples Vol. 85. Springer Science and Business Media (2012). | MR | Zbl
[26] , Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, N.J. (1964). | MR | Zbl
[27] and , Adaptive nonparametric drift estimation of an integrated jump diffusion process. ESAIM: PS 22 (2018) 236–260. | MR | Numdam | DOI
[28] , Stability of differential equations. Sijthoff and Noordhoff, Germantown, MD (1980). | DOI
[29] , and , Non-parametric estimation of the death rate in branching diffusions. Scand. J. Stat. 29 (2002) 665–692. | MR | Zbl | DOI
[30] and , On nonparametric tests of positivity/monotonicity/convexity. Ann. Stat. (2002) 498–527. | MR | Zbl
[31] , A jump-diffusion model for option pricing. Manag. Sci. 48 (2002) 1086–1101. | Zbl | DOI
[32] , Some problems of nonparametric estimation by observations of ergodic diffusion process. Statist. Probab. Lett. 32 (1997) 311–320. | MR | Zbl | DOI
[33] and , Recursive computation of the invariant distribution of a diffusion. Bernoulli 8 (2002) 367–405. | MR | Zbl
[34] , Density estimation for a class of continuous time processes. Math. Methods Stat. 6 (1997) 171–199. | MR | Zbl
[35] and , Optimal pointwise adaptive methods in nonparametric estimation. Ann. Stat. (1997) 2512–2546. | MR | Zbl
[36] , Ergodicity and exponential beta - mixing bounds for multidimensional diffusions with jumps. Stoch. Process. Appl. 117 (2007) 35–56. | MR | Zbl | DOI
[37] , Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3 (1976) 125–144. | Zbl | DOI
[38] , Density estimation in a continuous-time Markov processes. Ann. Statist. 7 (1979) 341–348. | MR | Zbl | DOI
[39] , Recursive computation of the invariant measure of a stochastic differential equation driven by a Lévy process. Ann. Appl. Probab. 18 (2008) 379–426. | MR | Zbl | DOI
[40] , Non parametric estimation of the diffusion coefficients of a diffusion with jumps. Stoch. Process. Appl. 129 (2019) 5364–5405. | MR | DOI
[41] , Adaptive invariant density estimation for ergodic diffusions over anisotropic classes. Ann. Stat. 46 (2018) 3451–3480. | MR | DOI
[42] , On nonparametric estimation of density level sets. Ann. Stat. 25 (1997) 948–969. | MR | Zbl | DOI
[43] , Introduction to nonparametric estimation. Springer Science and Business Media (2008). | MR | Zbl
[44] , Rates of convergence and asymptotic normality of kernel estimators for ergodic diffusion processes. Nonparametric Statist. 13 (2001) 833–850. | MR | Zbl | DOI
[45] , On Castellana-Leadbetter’s condition for diffusion density estimation. Stat. Inference Stoch. Process. 2 (1999) 1–9. | MR | Zbl | DOI
Cité par Sources :
