## Bernoulli

• Bernoulli
• Volume 23, Number 3 (2017), 1874-1910.

### Efficient estimation for diffusions sampled at high frequency over a fixed time interval

#### Abstract

Parametric estimation for diffusion processes is considered for high frequency observations over a fixed time interval. The processes solve stochastic differential equations with an unknown parameter in the diffusion coefficient. We find easily verified conditions on approximate martingale estimating functions under which estimators are consistent, rate optimal, and efficient under high frequency (in-fill) asymptotics. The asymptotic distributions of the estimators are shown to be normal variance-mixtures, where the mixing distribution generally depends on the full sample path of the diffusion process over the observation time interval. Utilising the concept of stable convergence, we also obtain the more easily applicable result that for a suitable data dependent normalisation, the estimators converge in distribution to a standard normal distribution. The theory is illustrated by a simulation study comparing an efficient and a non-efficient estimating function for an ergodic and a non-ergodic model.

#### Article information

Source
Bernoulli, Volume 23, Number 3 (2017), 1874-1910.

Dates
Revised: November 2015
First available in Project Euclid: 17 March 2017

https://projecteuclid.org/euclid.bj/1489737628

Digital Object Identifier
doi:10.3150/15-BEJ799

Mathematical Reviews number (MathSciNet)
MR3624881

Zentralblatt MATH identifier
06714322

#### Citation

Jakobsen, Nina Munkholt; Sørensen, Michael. Efficient estimation for diffusions sampled at high frequency over a fixed time interval. Bernoulli 23 (2017), no. 3, 1874--1910. doi:10.3150/15-BEJ799. https://projecteuclid.org/euclid.bj/1489737628

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