Continuous and Discrete Time Modeling

Abstract

As mentioned, the transition of a continuous time model into a discrete time model is not an easy issue. We discuss here various discretization procedures to turn continuous time into discrete time models. There are many methods to convert continuous time models into discrete time variants. The main discretization methods are the Euler method, the Milstein method and a new local linearization method. All those will be illustrated here to obtain discrete-time approximate models.

Appendix

7.1.1 Tables: Estimation Results

Table 7.1 Results of estimation and forecast for Germany^a

Table 7.2 Results of estimation and forecast for the U.K.^a

Table 7.3 Results of estimation and forecast for the U.S.^a

7.1.2 The Likelihood Function of the Milstein Approximation

Here we show the derivation of the likelihood function when using the Milstein method. Following (7.3), the dynamic of the SDE (7.1) is approximated by

$$\displaystyle{ X_{t_{i+1}} - X_{t_{i}} = (c -\beta X_{t_{i}})\varDelta t_{i} +\sigma X_{t_{i}}^{\gamma }\varDelta W_{t_{i}} + \frac{1} {2}\sigma ^{2}\gamma X_{ t_{i}}^{2\gamma -1}(\varDelta W_{ t_{i}}^{2} -\varDelta t_{ i}), }$$

(7.14)

where $\varDelta t_{i} = t_{i+1} - t_{i}$, $\varDelta W_{t_{i}} = W_{t_{i+1}} - W_{t_{i}}$. Let

$$\displaystyle{Y _{t_{i+1}} = X_{t_{i+1}} - X_{t_{i}} - (c -\beta X_{t_{i}})\varDelta t_{i} + \frac{1} {2}\sigma ^{2}\gamma X_{ t_{i}}^{2\gamma -1}\varDelta t_{ i}.}$$

Then (7.14) becomes

$$\displaystyle{ \frac{1} {2}\sigma ^{2}\gamma X_{ t_{i}}^{2\gamma -1}(\varDelta W_{ t_{i}})^{2} +\sigma X_{ t_{i}}^{\gamma }\varDelta W_{t_{i}} = Y _{t_{i+1}}. }$$

(7.15)

Let $x_{i} \in \mathbb{R}$ still be the realizations of $X_{t_{i}}$ and y _i be the realizations of $Y _{t_{i}}$ for i = 0, …, N correspondingly. We solve Eq. (7.15) to obtain the realizations of $\varDelta W_{t_{i}} = u_{i+1}^{+},u_{i+1}^{-}$, where

$$\displaystyle{\begin{array}{rl} u_{i+1}^{+} & = \frac{-1+\sqrt{1+\frac{2\gamma y_{i+1 } } {x_{i}}} } {\sigma \gamma x_{i}^{\gamma -1}} \\ u_{i+1}^{-}& = \frac{-1-\sqrt{1+\frac{2\gamma y_{i+1 } } {x_{i}}} } {\sigma \gamma x_{i}^{\gamma -1}}.\end{array} }$$

Then the conditional density is given by

$$\displaystyle{ \begin{array}{rl} &p\big(X_{t_{i+1}} = x_{i+1}\big\vert X_{t_{i}} = x_{i}\big) = \frac{dP\big(\{\varDelta W_{t_{i}}=du_{i+1}^{+}\}\cup \{\varDelta W_{ t_{i}}=du_{i+1}^{-}\}\big)} {dy_{i+1}} \\ =&\frac{dP\big(\varDelta W_{t_{i}}=du_{i+1}^{+}\big)} {du_{i+1}^{+}} \bigg\vert \frac{du_{i+1}^{+}} {dy_{i+1}} \bigg\vert + \frac{dP\big(\varDelta W_{t_{i}}=du_{i+1}^{-}\big)} {du_{i+1}^{-}} \bigg\vert \frac{du_{i+1}^{-}} {dy_{i+1}} \bigg\vert \\ =& \frac{1} {\sqrt{2\pi \varDelta t_{i}}}\bigg(\exp \big(-\frac{(u_{i+1}^{+})^{2}} {2\varDelta t_{i}} \big) +\exp \big (-\frac{(u_{i+1}^{-})^{2}} {2\varDelta t_{i}} \big)\bigg)\bigg\vert \frac{1} {\sigma x_{i}^{\gamma }\sqrt{1+\frac{2\gamma y_{i+1 } } {x_{i}}} }\bigg\vert, \end{array} }$$

(7.16)

as $1 + \frac{2\gamma y_{i+1}} {x_{i}}> 0$. If $1 + \frac{2\gamma y_{i+1}} {x_{i}} <0$, then the density above is infinity. If $1 + \frac{2\gamma y_{i+1}} {x_{i}} <0$, which means there is no real solution of $\varDelta W_{t_{i}}$ in (7.15) for such y _i+1, therefore the density is equal to zero

$$\displaystyle{p\big(X_{t_{i+1}} = dx_{i+1}\big\vert X_{t_{i}} = x_{i}\big) = 0.}$$

Comparing the density function (7.16) and the density function in (2.5) p. 7 in Elerian (1998), it is not difficult to show the identity of these two functions by some calculation. By numerical operations of the ML estimations we must modify the density function, because when $1 + \frac{2\gamma y_{i+1}} {x_{i}} = 0$, the value of the density function is infinity. Therefore we apply the following density function for the ML estimations:

$$\displaystyle\begin{array}{rcl} & & g_{mil}(x_{i},x_{i+1},\theta,\varDelta t_{i}) = \frac{dP\big(X_{t_{i+1}} = dx_{i+1}\big\vert X_{t_{i}} = x_{i}\big)} {dx_{i+1}} {}\\ & & = \frac{1} {\sqrt{2\pi \varDelta t_{i}}}\bigg(\exp \big(-\frac{(u_{i+1}^{+})^{2}} {2\varDelta t_{i}} \big) +\exp \big (-\frac{(u_{i+1}^{-})^{2}} {2\varDelta t_{i}} \big)\bigg)\bigg\vert \frac{1} {\sigma x_{i}^{\gamma }\sqrt{1 + \frac{2\gamma y_{i+1 } } {x_{i}}} }\bigg\vert, {}\\ & & \mbox{ for}\,1 + \frac{2\gamma y_{i+1}} {x_{i}}> 10^{-10} {}\\ & & {}\\ & & = 10^{-10}, {}\\ & & \mbox{ otherwise}. {}\\ \end{array}$$