 
Summary: The Annals of Statistics
2004, Vol. 32, No. 5, 21862222
DOI 10.1214/009053604000000427
© Institute of Mathematical Statistics, 2004
ESTIMATORS OF DIFFUSIONS WITH RANDOMLY SPACED
DISCRETE OBSERVATIONS: A GENERAL THEORY
BY YACINE AÏTSAHALIA1 AND PER A. MYKLAND2
Princeton University and University of Chicago
We provide a general method to analyze the asymptotic properties
of a variety of estimators of continuous time diffusion processes when
the data are not only discretely sampled in time but the time separating
successive observations may possibly be random. We introduce a new
operator, the generalized infinitesimal generator, to obtain Taylor expansions
of the asymptotic moments of the estimators. As a special case, our results
apply to the situation where the data are discretely sampled at a fixed
nonrandom time interval. We include as specific examples estimators based
on maximumlikelihood and discrete approximations such as the Euler
scheme.
1. Introduction. Most theoretical models in finance are spelled out in
continuous time [see, e.g., Merton (1992)], whereas the observed data are, by
