 
Summary: The Annals of Statistics
2008, Vol. 36, No. 2, 906937
DOI: 10.1214/009053607000000622
© Institute of Mathematical Statistics, 2008
CLOSEDFORM LIKELIHOOD EXPANSIONS
FOR MULTIVARIATE DIFFUSIONS
BY YACINE AÏTSAHALIA1
Princeton University
This paper provides closedform expansions for the loglikelihood func
tion of multivariate diffusions sampled at discrete time intervals. The coef
ficients of the expansion are calculated explicitly by exploiting the special
structure afforded by the diffusion model. Examples of interest in financial
statistics and Monte Carlo evidence are included, along with the convergence
of the expansion to the true likelihood function.
1. Introduction. Diffusions and, more generally, continuoustime Markov
processes are generally specified in economics and finance by their evolution over
infinitesimal instants, that is, by writing down the stochastic differential equation
followed by the state vector. However, for most estimation strategies relying on
discretely sampled data, we need to be able to infer the implications of the in
finitesimal time evolution of the process for the finite time intervals at which the
