12.1 Introduction
Let us assume a strictly stationary one-dimensional
diffusion
solving the
stochastic differential equation (SDE)
 |
|
|
(12.1) |
where the driving process
in (12.1) is a standard Wiener process.
In a mathematical finance setting,
might be the price process
of a stock, a stock market index or any other observable process.
For the rest of the chapter the
drift
,
and the diffusion coefficient
in (12.1)
are assumed to be sufficiently smooth, so that a unique solution
of (12.1) exists.
In applications we are mostly interested in the stationary
solutions of (12.1).
For the existence of a stationary solution, the drift and
the diffusion coefficient must satisfy some conditions,
Bibby and Sørensen (1995).
The most important condition is that the stationary forward
Kolmogorov equation
has a solution
which is a probability density.
If the initial value
is distributed in accordance with
, and if it is independent of the Wiener process
in (12.1), then (12.1) defines a stationary process.
The above condition holds for the Ornstein-Uhlenbeck
process with a normal stationary distribution,
and for the Cox-Ingersoll-Ross process with a
-distribution.
For the statistical analysis we assume that
is observed at
discrete times
,
with a time step size
.
From these observations we get a time series
with
certain dynamics specified in Section 12.2.
The aim of this chapter is to test a parametric model for the drift
function
against a nonparametric alternative, i.e.
 |
(12.2) |
where
is an unknown parameter.
The test statistic we apply is based on the empirical likelihood.
This concept was introduced by Chen et al. (2001) for time series.
To apply it in our situation we start with the discretization
of the diffusion process
.