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

Nina Munkholt Jakobsen, Michael Sørensen

Research output: Contribution to journalJournal articleResearchpeer-review


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.

Original languageEnglish
Issue number3
Pages (from-to)1874-1910
Number of pages37
Publication statusPublished - 1 Aug 2017
Externally publishedYes


  • Approximate martingale estimating functions
  • Discrete time sampling of diffusions
  • In-fill asymptotics
  • Normal variance-mixtures
  • Optimal rate
  • Random Fisher information
  • Stable convergence
  • Stochastic differential equation


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