12.1 Introduction

Let us assume a strictly stationary one-dimensional diffusion $ Z$ solving the stochastic differential equation (SDE)

$\displaystyle dZ(t) = m\{Z(t)\} dt + v\{Z(t)\} dW(t)$     (12.1)

where the driving process $ W = \textrm{$\{ W(t), \; t \in [0, \infty) \}$}$ in (12.1) is a standard Wiener process. In a mathematical finance setting, $ Z$ 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 $ m: \mathbb{R}\mapsto \mathbb{R}$, and the diffusion coefficient $ v: \mathbb{R}\mapsto [0, \infty)$ 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

$\displaystyle (1/2)\,\big\{ v^2(z)p(z)\, \big \}'-m(z)p(z)=0$

has a solution $ p(z)$ which is a probability density. If the initial value $ Z(0)$ is distributed in accordance with $ p_0$, and if it is independent of the Wiener process $ W(t)$ 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 $ \Gamma$-distribution. For the statistical analysis we assume that $ Z$ is observed at discrete times $ t_i = i\Delta,\, i=1,2,\dots , n$, with a time step size $ \Delta > 0$. From these observations we get a time series $ Z^{\Delta}$ with certain dynamics specified in Section 12.2.

The aim of this chapter is to test a parametric model for the drift function $ m$ against a nonparametric alternative, i.e.

$\displaystyle H_0(m) : m(z) = m_\theta(z)$ (12.2)

where $ \theta $ 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 $ Z$.