## The Annals of Statistics

### Jump filtering and efficient drift estimation for Lévy-driven SDEs

#### Abstract

The problem of drift estimation for the solution $X$ of a stochastic differential equation with Lévy-type jumps is considered under discrete high-frequency observations with a growing observation window. An efficient and asymptotically normal estimator for the drift parameter is constructed under minimal conditions on the jump behavior and the sampling scheme. In the case of a bounded jump measure density, these conditions reduce to $n\Delta_{n}^{3-\varepsilon}\rightarrow 0$, where $n$ is the number of observations and $\Delta_{n}$ is the maximal sampling step. This result relaxes the condition $n\Delta_{n}^{2}\rightarrow 0$ usually required for joint estimation of drift and diffusion coefficient for SDEs with jumps. The main challenge in this estimation problem stems from the appearance of the unobserved continuous part $X^{c}$ in the likelihood function. In order to construct the drift estimator, we recover this continuous part from discrete observations. More precisely, we estimate, in a nonparametric way, stochastic integrals with respect to $X^{c}$. Convergence results of independent interest are proved for these nonparametric estimators.

#### Article information

Source
Ann. Statist., Volume 46, Number 4 (2018), 1445-1480.

Dates
Revised: May 2017
First available in Project Euclid: 27 June 2018

https://projecteuclid.org/euclid.aos/1530086422

Digital Object Identifier
doi:10.1214/17-AOS1591

Mathematical Reviews number (MathSciNet)
MR3819106

Zentralblatt MATH identifier
06936467

#### Citation

Gloter, Arnaud; Loukianova, Dasha; Mai, Hilmar. Jump filtering and efficient drift estimation for Lévy-driven SDEs. Ann. Statist. 46 (2018), no. 4, 1445--1480. doi:10.1214/17-AOS1591. https://projecteuclid.org/euclid.aos/1530086422

#### References

• [1] Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge Univ. Press, Cambridge.
• [2] Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc. Ser. B. Stat. Methodol. 63 167–241.
• [3] Bibinger, M. and Winkelmann, L. (2015). Econometrics of co-jumps in high-frequency data with noise. J. Econometrics 184 361–378.
• [4] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton, FL.
• [5] Ditlevsen, S. and Greenwood, P. (2013). The Morris-Lecar neuron model embeds a leaky integrate-and-fire model. J. Math. Biol. 67 239–259.
• [6] Figueroa-López, J. E. and Nisen, J. (2013). Optimally thresholded realized power variations for Lévy jump diffusion models. Stochastic Process. Appl. 123 2648–2677.
• [7] Florens-Zmirou, D. (1989). Approximate discrete-time schemes for statistics of diffusion processes. Statistics 20 547–557.
• [8] Genon-Catalot, V. and Jacod, J. (1993). On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Ann. Inst. Henri Poincaré Probab. Stat. 29 119–151.
• [9] Gloter, A., Loukianova, D. and Mai, H. (2018). Supplement to “Jump filtering and efficient drift estimation for Lévy-driven SDEs.” DOI:10.1214/17-AOS1591SUPP.
• [10] Hutton, J. E. and Nelson, P. I. (1984). Interchanging the order of differentiation and stochastic integration. Stochastic Process. Appl. 18 371–377.
• [11] Ibragimov, I. and Has’minskii, R. (2013). Statistical Estimation: Asymptotic Theory, Springer-Verlag, New York.
• [12] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin.
• [13] Kessler, M. (1997). Estimation of an ergodic diffusion from discrete observations. Scand. J. Stat. 24 211–229.
• [14] Kou, S. G. (2002). A jump-diffusion model for option pricing. Manage. Sci. 48 1086–1101.
• [15] Küchler, U. and Sørensen, M. (1999). A note on limit theorems for multivariate martingales. Bernoulli 5 483–493.
• [16] Loukianova, D. and Loukianov, O. (2005). Uniform law of large numbers and consistency of estimators for Harris diffusions. Statist. Probab. Lett. 74 347–355.
• [17] Mai, H. (2014). Efficient maximum likelihood estimation for Lévy-driven Ornstein-Uhlenbeck processes. Bernoulli 20 919–957.
• [18] Mancini, C. (2011). The speed of convergence of the threshold estimator of integrated variance. Stochastic Process. Appl. 121 845–855.
• [19] Masuda, H. (2007). Ergodicity and exponential $\beta$-mixing bounds for multidimensional diffusions with jumps. Stochastic Process. Appl. 117 35–56.
• [20] Masuda, H. (2009). Erratum to: “Ergodicity and exponential $\beta$-mixing bound for multidimensional diffusions with jumps” [Stochastic Process. Appl. 117 (2007) 35–56] [MR2287102]. Stochastic Process. Appl. 119 676–678.
• [21] Masuda, H. (2010). Approximate self-weighted LAD estimation of discretely observed ergodic Ornstein-Uhlenbeck processes. Electron. J. Stat. 4 525–565.
• [22] Masuda, H. (2013). Convergence of Gaussian quasi-likelihood random fields for ergodic Lévy driven SDE observed at high frequency. Ann. Statist. 41 1593–1641.
• [23] Merton, R. (1976). Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3 125–144.
• [24] Ogihara, T. and Yoshida, N. (2011). Quasi-likelihood analysis for the stochastic differential equation with jumps. Stat. Inference Stoch. Process. 14 189–229.
• [25] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin.
• [26] Shimizu, Y. (2006). $M$-estimation for discretely observed ergodic diffusion processes with infinitely many jumps. Stat. Inference Stoch. Process. 9 179–225.
• [27] Shimizu, Y. (2008). Some remarks on estimation of diffusion coefficients for jump-diffusions from finite samples. Bull. Inform. Cybernet. 40 51–60.
• [28] Shimizu, Y. (2008). A practical inference for discretely observed jump-diffusions from finite samples. J. Japan Statist. Soc. 38 391–413.
• [29] Shimizu, Y. and Yoshida, N. (2006). Estimation of parameters for diffusion processes with jumps from discrete observations. Stat. Inference Stoch. Process. 9 227–277.
• [30] Tran, N. K. (2014). LAN property for jump diffusion processes with discrete observations via Malliavin calculus Ph.D. thesis Univ. Paris 13.
• [31] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge Univ. Press, Cambridge.
• [32] Yoshida, N. (1992). Estimation for diffusion processes from discrete observation. J. Multivariate Anal. 41 220–242.

#### Supplemental materials

• Jump filtering and efficient drift estimation for Lévy-driven SDEs. The supplement contains the two additional Sections 7–8. In Section 7, we investigate the numerical performance of the estimator for the finite sample. We consider the case of Ornstein–Uhlbenbeck and “Hyperbolic” diffusion models, with finite or infinite activity jump measure. We compare the results for different choices of the threshold constants $a_{i}^{n}$ [recall (3.3)]. In Section 8, we give a proof the Lemma 2.1 about the ergodicity of the process, and of the LAN property (Theorem 5.3). Next, we gather the proofs of the technical Lemmas 6.1, 6.3 and 6.4, and show that the identifiability Assumption 6 is equivalent to (2.1).