## Bernoulli

• Bernoulli
• Volume 20, Number 3 (2014), 1059-1096.

### Asymptotic lower bounds in estimating jumps

#### Abstract

We study the problem of the efficient estimation of the jumps for stochastic processes. We assume that the stochastic jump process $(X_{t})_{t\in[0,1]}$ is observed discretely, with a sampling step of size $1/n$. In the spirit of Hajek’s convolution theorem, we show some lower bounds for the estimation error of the sequence of the jumps $(\Delta X_{T_{k}})_{k}$. As an intermediate result, we prove a LAMN property, with rate $\sqrt{n}$, when the marks of the underlying jump component are deterministic. We deduce then a convolution theorem, with an explicit asymptotic minimal variance, in the case where the marks of the jump component are random. To prove that this lower bound is optimal, we show that a threshold estimator of the sequence of jumps $(\Delta X_{T_{k}})_{k}$ based on the discrete observations, reaches the minimal variance of the previous convolution theorem.

#### Article information

Source
Bernoulli, Volume 20, Number 3 (2014), 1059-1096.

Dates
First available in Project Euclid: 11 June 2014

https://projecteuclid.org/euclid.bj/1402488934

Digital Object Identifier
doi:10.3150/13-BEJ515

Mathematical Reviews number (MathSciNet)
MR3217438

Zentralblatt MATH identifier
06327903

#### Citation

Clément, Emmanuelle; Delattre, Sylvain; Gloter, Arnaud. Asymptotic lower bounds in estimating jumps. Bernoulli 20 (2014), no. 3, 1059--1096. doi:10.3150/13-BEJ515. https://projecteuclid.org/euclid.bj/1402488934

#### References

• [1] Aït-Sahalia, Y. (2002). Telling from discrete data whether the underlying continuous-time model is a diffusion. J. Finance 57 2075–2112.
• [2] Aït-Sahalia, Y., Fan, J. and Peng, H. (2009). Nonparametric transition-based tests for jump diffusions. J. Amer. Statist. Assoc. 104 1102–1116.
• [3] Aït-Sahalia, Y. and Jacod, J. (2009). Estimating the degree of activity of jumps in high frequency data. Ann. Statist. 37 2202–2244.
• [4] Aït-Sahalia, Y. and Jacod, J. (2009). Testing for jumps in a discretely observed process. Ann. Statist. 37 184–222.
• [5] Azencott, R. (1984). Densité des diffusions en temps petit: Développements asymptotiques. I. In Seminar on Probability, XVIII. Lecture Notes in Math. 1059 402–498. Berlin: Springer.
• [6] Barndorff-Nielsen, O.E. and Shephard, N. (2006). Econometrics of testing for jumps in financial economics using bipower variation. J. Financial Econometrics 4 1–30.
• [7] Barndorff-Nielsen, O.E., Shephard, N. and Winkel, M. (2006). Limit theorems for multipower variation in the presence of jumps. Stochastic Process. Appl. 116 796–806.
• [8] Cont, R. and Mancini, C. (2011). Nonparametric tests for pathwise properties of semimartingales. Bernoulli 17 781–813.
• [9] Gloter, A. and Gobet, E. (2008). LAMN property for hidden processes: The case of integrated diffusions. Ann. Inst. Henri Poincaré Probab. Stat. 44 104–128.
• [10] Gobet, E. (2001). Local asymptotic mixed normality property for elliptic diffusion: A Malliavin calculus approach. Bernoulli 7 899–912.
• [11] Huang, X. and Tauchen, G. (2006). The relative contribution of jumps of total price variance. J. Financial Econometrics 4 456–499.
• [12] Ibragimov, I.A. and Has’minskiĭ, R.Z. (1981). Statistical Estimation: Asymptotic theory. Applications of Mathematics 16. New York: Springer. Translated from the Russian by Samuel Kotz.
• [13] Jacod, J. (2008). Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Process. Appl. 118 517–559.
• [14] Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307.
• [15] Jacod, J. and Todorov, V. (2010). Do price and volatility jump together? Ann. Appl. Probab. 20 1425–1469.
• [16] Jeganathan, P. (1981). On a decomposition of the limit distribution of a sequence of estimators. Sankhyā Ser. A 43 26–36.
• [17] Jeganathan, P. (1982). On the asymptotic theory of estimation when the limit of the log-likelihood ratios is mixed normal. Sankhyā Ser. A 44 173–212.
• [18] Kunita, H. (1997). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge: Cambridge Univ. Press. Reprint of the 1990 original.
• [19] Mancini, C. (2004). Estimation of the characteristics of the jumps of a general Poisson-diffusion model. Scand. Actuar. J. 1 42–52.
• [20] Mancini, C. (2009). Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps. Scand. J. Stat. 36 270–296.
• [21] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Probability and Its Applications (New York). Berlin: Springer.
• [22] van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge: Cambridge Univ. Press.
• [23] Woerner, J.H.C. (2006). Power and multipower variation: Inference for high frequency data. In Stochastic Finance 343–364. New York: Springer.