Adaptive bandwidth selection; anisotropic density estimation; ergodic diffusion with jumps
Résumé :
[en] We consider the solution of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density μ. We assume that a continuous record of observations is available.
In the case without jumps, Reiss and Dalalyan [7] and Strauch [24] have found convergence rates of invariant density estimators, under respectively isotropic and anisotropic H ̈older smoothness constraints, which are considerably faster than those known from standard multivariate density estimation.
We extend the previous works by obtaining, in presence of jumps, some estimators which have the same convergence rates they had in the case without jumps for d ≥ 2 and a rate which depends on the degree of the jumps in the one-dimensional setting.
We propose moreover a data driven bandwidth selection procedure based on the Goldenshluger and Lepski method [11] which leads us to an adaptive non-parametric kernel estimator of the stationary density μ of the jump diffusion X.
Disciplines :
Mathématiques
Auteur, co-auteur :
AMORINO, Chiara ✱; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
Gloter, Arnaud ✱; Université d'Evry > Lamme
✱ Ces auteurs ont contribué de façon équivalente à la publication.
Co-auteurs externes :
yes
Langue du document :
Anglais
Titre :
Invariant density adaptive estimation for ergodic jump diffusion processes over anisotropic classes
Applebaum, David, Lévy Processes and Stochastic Calculus. 2009, Cambridge university press.
Barndorff-Nielsen, O.E., Shephard, N., 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.
Bates, D.S., Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark. Rev. Financ. Stud. 9:1 (1996), 69–107.
Chen, Z.Q., Hu, E., Xie, L., Zhang, X., Heat kernels for non-symmetric diffusion operators with jumps. J. Differential Equations 263:10 (2017), 6576–6634.
Comte, F., Lacour, C., Anisotropic adaptive kernel deconvolution. Ann. Inst. Henri Poincaré Probab. Stat. 49:2 (2013), 569–609.
Comte, F., Merlevède, F., Adaptive estimation of the stationary density of discrete and continuous time mixing processes. ESAIM Probab. Stat. 6 (2002), 211–238.
Comte, F., Merlevède, F., Super optimal rates for nonparametric density estimation via projection estimators. Stochastic Process. Appl. 115:5 (2005), 797–826.
Comte, F., Prieur, C., Samson, A., Adaptive estimation for stochastic damping hamiltonian systems under partial observation. Stochastic Process. Appl. 127:11 (2017), 3689–3718.
Dalalyan, A., Reiss, M., Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case. Probab. Theory Related Fields 137:1 (2007), 25–47.
Ditlevsen, S., Greenwood, P., The Morris–Lecar neuron model embeds a leaky integrate-and-fire model. J. Math. Biol. 67 (2013), 239–259.
Doukhan, P., Mixing: Properties and Examples, Vol. 85. 2012, Springer Science and Business Media.
Funke, B., Schmisser, E., Adaptive nonparametric drift estimation of an integrated jump diffusion process. ESAIM Probab. Stat. 22 (2018), 236–260.
Goldenshluger, A., Lepski, O., Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality. Ann. Statist. 39:3 (2011), 1608–1632.
Klein, T., Rio, E., Concentration around the mean for maxima of empirical processes. Ann. Probab. 33:3 (2005), 1060–1077.
Kou, S.G., A jump-diffusion model for option pricing. Manage. Sci. 48 (2002), 1086–1101.
Kusuoka, S., Yoshida, N., Malliavin calculus, geometric mixing, and expansion of diffusion functionals. Probab. Theory Related Fields 116:4 (2000), 457–484.
Kutoyants, Y.A., Statistical Inference for Ergodic Diffusion Processes. 2013, Springer Science and Business Media.
Lacour, C., Massart, P., Rivoirard, V., Estimator selection: a new method with applications to kernel density estimation. Sankhya A 79:2 (2017), 298–335.
Lepski, O., Multivariate density estimation under sup-norm loss: oracle approach, adaptation and independence structure. Ann. Statist. 41:2 (2013), 1005–1034.
Masuda, H., Ergodicity and exponential beta - mixing bounds for multidimensional diffusions with jumps. Stochastic Process. Appl. 117:1 (2007), 35–56.
Merton, R.C., Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3 (1976), 125–144.
Meyn, S.P., Tweedie, R.L., Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 25:3 (1993), 518–548.
Nikolskii, S.M., Approximation of Functions of Several Variables and Embedding Theorems. 1975, Springer, Berlin.
Schmisser, E., Nonparametric estimation of the derivatives of the stationary density for stationary processes. ESAIM Probab. Stat. 17 (2013), 33–69.
Schmisser, E., Non parametric estimation of the diffusion coefficients of a diffusion with jumps. Stochastic Process. Appl. 129:12 (2019), 5364–5405.
Stramer, O., Tweedie, R.L., Existence and stability of weak solutions to stochastic differential equations with non-smooth coefficients. Statist. Sinica 57 (1997), 7–593.
Strauch, C., Adaptive invariant density estimation for ergodic diffusions over anisotropic classes. Ann. Statist. 46:6B (2018), 3451–3480.
Tsybakov, A.B., Introduction to Nonparametric Estimation. 2008, Springer Science & Business Media.
Veretennikov, A.Y., Bounds for the mixing rate in the theory of stochastic equations. Theory Probab. Appl. 32:2 (1988), 273–281 ISO 690.
Viennet, G., Inequalities for absolutely regular sequences: application to density estimation. Probab. Theory Related Fields 107:4 (1997), 467–492.
