## Bernoulli

• Bernoulli
• Volume 12, Number 3 (2006), 383-401.

### Parametric inference for discretely observed non-ergodic diffusions

Jean Jacod

#### Abstract

We consider a multidimensional diffusion process $X$ whose drift and diffusion coefficients depend respectively on a parameter $λ$ and $θ$. This process is observed at $n +1$ equally spaced times $0 ,Δ n,2Δ n,…,nΔ n$, and $T n =nΔ n$ denotes the length of the `observation window'. We are interested in estimating $λ$ and/or $θ$. Under suitable smoothness and identifiability conditions, we exhibit estimators $λ ̂ n$ and $θ ̂ n$, such that the variables $n .(θ ̂ n-θ)$ and $T n (λ ̂ n-λ)$ are tight for $Δ n →0$ and $T n →∞$. When $λ$ is known, we can even drop the assumption that $T n →∞$. These results hold without any kind of ergodicity or even recurrence assumption on the diffusion process.

#### Article information

Source
Bernoulli, Volume 12, Number 3 (2006), 383-401.

Dates
First available in Project Euclid: 28 June 2006

https://projecteuclid.org/euclid.bj/1151525127

Digital Object Identifier
doi:10.3150/bj/1151525127

Mathematical Reviews number (MathSciNet)
MR2232724

Zentralblatt MATH identifier
1100.62081

#### Citation

Jacod, Jean. Parametric inference for discretely observed non-ergodic diffusions. Bernoulli 12 (2006), no. 3, 383--401. doi:10.3150/bj/1151525127. https://projecteuclid.org/euclid.bj/1151525127

#### References

• [1] Aït-Sahalia, Y. (2002) Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach. Econometrica, 70, 223-262.
• [2] Basawa, I.V. and Scott, D.J. (1983) Asymptotic Optimal Inference for Non-ergodic Models, Lecture Notes in Statist. 17. New York: Springer-Verlag.
• [3] Dohnal, G. (1987) On estimating the diffusion coefficient. J. Appl. Probab., 24, 105-114.
• [4] Genon-Catalot, V. and Jacod, J. (1993) On the estimation of the diffusion coefficient for multidimensional diffusion processes. Ann. Inst. H. Poincaré Probab. Statist., 29, 119-151.
• [5] Ibragimov, I.A. and Has´minskii, R.Z. (1981) Statistical Estimation: Asymptotic Theory. New York: Springer-Verlag.
• [6] Kessler, M. (1997) Estimation of an ergodic diffusion from discrete observations. Scand. J. Statist., 24, 211-229.
• [7] Kessler, M. and Sørensen, M. (1999) Estimating equations based on eigenfunctions for a discretely observed diffusion process. Bernoulli, 5, 299-314.
• [8] Kutoyants, Yu.A. (2004) Statistical Inference for Ergodic Diffusion Processes. London: Springer- Verlag.
• [9] Pedersen, A.R. (1995) A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scand. J. Statist., 22, 55-71.
• [10] Prakasa Rao, B.L.S. (1999a) Semimartingales and Their Statistical Inference. Boca Raton, FL: Chapman & Hall.
• [11] Prakasa Rao, B.L.S. (1999b) Statistical Inference for Diffusion Type Processes. London: Arnold.
• [12] Protter, P. (1990) Stochastic Integration and Differential Equations. Berlin: Springer-Verlag.
• [13] Revuz, D. and Yor, M. (1991) Continuous Martingales and Brownian Motion. Berlin: Springer-Verlag.
• [14] Yoshida, N. (1992) Estimation for diffusion processes from discrete observation. J. Multivariate Anal., 41, 220-242.