## The Annals of Statistics

- Ann. Statist.
- Volume 18, Number 3 (1990), 1158-1171.

### A Sufficient Condition for Asymptotic Sufficiency of Incomplete Observations of a Diffusion Process

#### Abstract

Consider an $m$-dimensional diffusion process $(X_t)$ with unknown drift and small known variance observed on a time interval $\lbrack 0, T\rbrack$. We derive here a general condition ensuring the asymptotic sufficiency, in the sense of Le Cam, of incomplete observations of $(X_t)_{0 \leq t \leq T}$ with respect to the complete observation of the diffusion as the variance goes to 0. We then construct estimators based on these partial observations which are consistent, asymptotically Gaussian and asymptotically equivalent to the maximum likelihood estimator based on the observation of the complete sample path on $\lbrack 0, T\rbrack$. Finally, we study when this condition is satisfied for various incomplete observations which often arise in practice: discrete observations, observations of a smoothed diffusion, observation of the first hitting times and positions of concentric spheres, complete or partial observation of the record process for one-dimensional diffusions.

#### Article information

**Source**

Ann. Statist., Volume 18, Number 3 (1990), 1158-1171.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176347744

**Digital Object Identifier**

doi:10.1214/aos/1176347744

**Mathematical Reviews number (MathSciNet)**

MR1062703

**Zentralblatt MATH identifier**

0725.62073

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62B15: Theory of statistical experiments

Secondary: 62M05: Markov processes: estimation

**Keywords**

Diffusion process incomplete observations asymptotic sufficiency variance asymptotics parametric inference

#### Citation

Laredo, Catherine F. A Sufficient Condition for Asymptotic Sufficiency of Incomplete Observations of a Diffusion Process. Ann. Statist. 18 (1990), no. 3, 1158--1171. doi:10.1214/aos/1176347744. https://projecteuclid.org/euclid.aos/1176347744