Page 109 - Contributed Paper Session (CPS) - Volume 2

CPS1443 Shogo H.N. et al.
Although the contaminating noise has small variance such as 10 −4 , we
2
can see the serious influence on the performance of the existent method (LGA)
for the noisy diffusion model, and our proposal overwhelms in terms of both
mean and RMSE compared to LGA.

4. Discussion and Conclusion
We propose the adaptive estimation procedure for discretely and noisily
observed ergodic diffusion processes and show the theoretical results such as
consistency, asymptotic normality and convergence of moments of the
estimators. Our method has advantages such as asymptotic normality without
assuming that the variance of noise is known, convergence of moments
discussed in Yoshida (2011), and smaller computational burden for high-
dimensional parameters in optimisation compared to the existent
simultaneous approach. In simulation, we see unstable behaviour of the
estimator based on the LGA not considering the noise and the significance to
use our method when the data is contaminated by exogenous noise.

References
1. Adams, R. A. and Fournier, J. J. F. (2003). Sobolev spaces. Second edition.
Elsevier/Academic Press, Amsterdam.
2. Bibby, B. M. and Sørensen, M. (1995). Martingale estimating functions
for discretely observed diffusion processes. Bernoulli, 1:17–39.
3. Favetto, B. (2014). Parameter estimation by contrast minimization for
noisy observations of a diffusion process. Statistics, 48(6):1344–1370.
4. Favetto, B. (2016). Estimating functions for noisy observations of ergodic
diffusions. Statistical Inference for Stochastic Processes, 19:1–28.
5. Florens-Zmirou, D. (1989). Approximate discrete time schemes for
statistics of diffusion processes. Statistics, 20(4):547–557.
6. Gloter, A. and Jacod, J. (2001a). Diffusions with measurement errors. I.
local asymptotic normality. ESAIM: Probability and Statistics, 5:225–242.
7. Gloter, A. and Jacod, J. (2001b). Diffusions with measurement errors. II.
optimal estimators. ESAIM: Probability and Statistics, 5:243–260.
8. Jacod, J., Li, Y., Mykland, P. A., Podolskij, M., and Vetter, M. (2009).
Microstructure noise in the continuous case: the pre-averaging
approach. Stochastic Processes and their Applications, 119(7):2249–
2276.
9. Kessler, M. (1995). Estimation des parametres d’une diffusion par des
contrastes corriges. Comptes rendus de l’Académie des sciences. Série 1,
Mathématique, 320(3):359–362.
10. Kessler, M. (1997). Estimation of an ergodic diffusion from discrete
observations. Scandinavian Journal of Statistics, 24:211–229.

98 | I S I W S C 2 0 1 9