Adaptive nonparametric drift estimation of an integrated jump diffusion process

Funke, Benedikt¹ ; Schmisser, Émeline¹ 

¹

ESAIM: Probability and Statistics, Tome 22 (2018), pp. 236-260.

Résumé

In the present article, we investigate nonparametric estimation of the unknown drift function $b$ in an integrated Lévy driven jump diffusion model. Our aim will be to estimate the drift on a compact set based on a high-frequency data sample.

Instead of observing the jump diffusion process $V$ itself, we observe a discrete and high-frequent sample of the integrated process

$$X_t:={\int }_0^t{V}_s\mathrm{d}s$$

Based on the available observations of $X_t$, we will construct an adaptive penalized least-squares estimate in order to compute an adaptive estimator of the corresponding drift function $b$. Under appropriate assumptions, we will bound the $L^2$-risk of our proposed estimator. Moreover, we study the behavior of the proposed estimator in various Monte Carlo simulation setups.

Détail

DOI : 10.1051/ps/2018005

Classification : 62M09, 62G08
Keywords: Adaptive estimation, integrated jump diffusion, drift estimation, model selection, mean square estimator

Bibliographie

T. Andersen, T. Bollerslev, F. Diebold and P. Labys, The distribution of exchange rate volatility. J. Am. Stat. Assoc. 91 (2001) 42–55. | MR | Zbl | DOI

D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edn. Part of Cambridge Studies in Advanced Mathematics. Cambridge, UK (2009). | MR | DOI

F. Baltazar-Larios and M. Sørensen, Maximum likelihood estimation for integrated diffusion processes, in Contemporary Quantitative Finance: Essays in Honor of Eckhard Platen, edited by C. Chiarella and A. Novikov. Springer Heidelberg, Germany (2010). | MR | Zbl | DOI

F. Bandi and T. Nguyen, On the functional estimation of jump-diffusion models. J. Econom. 116 (2003) 293–328. | MR | Zbl | DOI

F. Bandi and P. Phillips, Fully nonparametric estimation of scalar diffusion models. Econometrica 71 (2003) 241–243. | MR | Zbl | DOI

A. Barron, L. Birgé and P. Massart, Risk bounds for model selection via penalization. Probab. Theory Relat. Fields 113 (1999) 301–413. | MR | Zbl | DOI

T. Bollerslev and H. Zhou, Estimating stochastic volatility diffusion using conditional moments of integrated volatility. J. Econom. 109 (2002) 33–65. | MR | Zbl | DOI

J. Campbell, A. Lo and A. Mackinlay, The Econometrics of Financial Markets. Princeton University Press, Princeton (1997). | Zbl | DOI

F. Comte, V. Genon-Catalot and Y. Rozenholc, Penalized nonparametric mean square error estimation of the coefficients of diffusion processes. Bernoulli 13 (2007) 514–543. | MR | Zbl | DOI

F. Comte, V. Genon-Catalot and Y. Rozenholc, Nonparametric adaptive estimation for integrated diffusions. Stoch. Process. Appl. 119 (2009) 811–834. | MR | Zbl | DOI

R.A. Devore and G.G. Lorentz, Constructive Approximation. Vol. 303 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg (1993). | MR | Zbl

S. Ditlevsen and M. Sørensen, Inference for observations of integrated diffusion processes. Scand. J. Stat. 31 (2004) 417–429. | MR | Zbl | DOI

D. Florens-Zmirou, On estimating the diffusion coefficient from discrete observations. J. Appl. Probab. 30 (1993) 790–804. | MR | Zbl | DOI

A. Gloter, Discrete sampling of an integrated diffusion process and parameter estimation of the diffusion coefficient. ESAIM: PS 4 (2000) 205–227. | MR | Zbl | mathdoc-id | DOI

A. Gloter, Parameter estimation for a discretely observed integrated process. Scand. J. Stat. 33 (2006) 83–104. | MR | Zbl | DOI

A. Gloter and E. Gobet, LAMN property for hidden processes: the case of integrated diffusions. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008) 104–128. | MR | Zbl | mathdoc-id | DOI

M. Lefebvre, On the inverse of the first hitting time problem for bidimensional processes. J. Appl. Probab. 34 (1997) 610–622. | MR | Zbl | DOI

H. Masuda, Ergodicity and exponential β-mixing bounds for multidimensional diffusions with jumps. Stoch. Process. Appl. 117 (2007) 35–56. | MR | Zbl | DOI

Y. Meyer, Ondelettes et opérateurs. I, in Actualités Mathématiques. [Current Mathematical Topics]. Hermann, Paris (1990). | MR | Zbl

J. Nicolau, Nonparametric estimation of second-order stochastic differential equations. Econom. Theory 23 (2007) 880–898. | MR | Zbl | DOI

E. Schmisser, Nonparametric adaptive estimation of the drift for a jump diffusion process. Stoch. Process. Appl. 124 (2014) 883–914. | MR | Zbl | DOI

Y. Song, Nonparametric estimation for second-order diffusion models in high frequency data. Singap. Econ. Rev. 63 (2017) 1–31.

Y. Song and Z. Lin, Empirical likelihood inference for the second-order jump-diffusion model. Stat. Probab. Lett. 83 (2013) 184–195. | MR | Zbl | DOI

Y. Song, Z. Lin and H. Wang, Re-weighted nadaraya-watson estimation of second-order jump-diffusion model. J. Stat. Plan. Inference 143 (2013) 730–744. | MR | Zbl | DOI