12.2 Discrete Time Approximation of a Diffusion

Let us assume that the diffusion process $ Z$ is observed at discrete times $ t_i = i\Delta,\, i=1,2,\dots , n$, with a time step size $ \Delta > 0$. Here we suppose that $ \Delta$ is small or, more precisely, will tend to zero asymptotically. Under rather weak assumptions, see Kloeden and Platen (1999), on the functions $ m$ and $ v^2$, it can be shown that the Euler approximation

$\displaystyle Z^{\Delta}(t)=Z^{\Delta}(0)+\int_0^t m \big\{ Z^{\Delta}(t_{i_s}) \big\} ds \ + \int_0^t v \big\{ Z^{\Delta}(t_{i_s}) \big\} dW(s) \ $ (12.3)

with $ t_{i_s}=\max\{ t_i \, , \, t_i\leq s \}$, converges in a mean square sense to $ Z$ as $ \Delta \to 0$, i.e.,

$\displaystyle \lim_{\Delta \rightarrow 0} \textrm{E}( \sup_{0 \leq t \leq T} \vert Z^{\Delta}(t) - Z(t) \vert^2 ) = 0 ,~~T > 0.$ (12.4)

From now on, we assume that a discrete time approximation $ Z^{\Delta}$ exists in the form of (12.3), and that the property (12.4) holds. For the purposes of this chapter, $ \Delta$ will always be considered small enough that one can substitute $ Z$ by $ Z^{\Delta}$ in our interpretation of the observed data. The increments of the Euler approximation and so the observed data will have the form

$\displaystyle Z^{\Delta}(t_{i+1})-Z^{\Delta}(t_i) =
m \big\{ Z^{\Delta}(t_{i}) ...
...\} \Delta
\ + v \big\{ Z^{\Delta}(t_{i}) \big\} \big\{W(t_{i+1}) - W(t_i)\big\}$     (12.5)

for $ i=0,1,\dots\,.$. The observations $ \{Z^{\Delta}(t_{i})\},~i=0,1,\dots\, n$ form a time series. As long as the step size $ \Delta$ is small enough the concrete choice of $ \Delta$ does not matter since all the relevant information about the model is contained in the drift $ m$ and diffusion coefficient $ v$.

For the following we introduce the notation

$\displaystyle X_i$ $\displaystyle \stackrel{\mathrm{def}}{=}$ $\displaystyle Z^{\Delta}(t_{i}) \; , \qquad X \stackrel{\mathrm{def}}{=}(X_1, \ldots, X_n)$  
$\displaystyle \varepsilon_i$ $\displaystyle \stackrel{\mathrm{def}}{=}$ $\displaystyle W(t_{i+1}) - W(t_i) \; ,
\qquad \varepsilon \stackrel{\mathrm{def}}{=}(\varepsilon_1 , \ldots , \varepsilon_n)$  
$\displaystyle Y_i$ $\displaystyle \stackrel{\mathrm{def}}{=}$ $\displaystyle X_{i+1} - X_i
= m \big(X_i \big) \Delta
\ + v \big(X_i \big) \varepsilon_i \; , \qquad
Y \stackrel{\mathrm{def}}{=}(Y_1, \ldots, Y_n)$ (12.6)

We can now apply the empirical likelihood Goodness-of-Fit test for stationary time series developed by Chen et al. (2001).