Let us assume that the diffusion process is observed at discrete
times
,
with a time step size
.
Here we suppose that
is small
or, more precisely, will tend to zero asymptotically.
Under rather weak assumptions, see Kloeden and Platen (1999),
on the functions
and
, it can be shown that the
Euler approximation
From now on, we assume that a discrete time
approximation
exists in the form of
(12.3), and that the property (12.4) holds.
For the purposes of this chapter,
will always be considered small enough that
one can substitute
by
in our interpretation
of the observed data.
The increments of the Euler approximation and so the
observed data will have the form
For the following we introduce the notation
We can now apply the empirical likelihood Goodness-of-Fit test
for stationary time series developed by Chen et al. (2001).