The processes x_{t} = ln(S_{t }e^{r t}/_{}), v_{t}, t Î R^{+}, satisfy equations (1.7), (1.8) (1.9),
(1.10) and as a consequence the
processes x_{t} = ln(S_{t}),
v_{t}, tÎ R^{+}, satisfy the
equations:
_{}
_{}
with the
initial conditions:
_{}
_{}
Recall that _{} and that the parameters
m and e are respectively
the drift and the volatility of the variance process that appear in equation
(1.2). As already done in Section
1 the stochastic differentials dW_{t}, dZ_{t}, t Î R^{+} are assumed to
satisfy the condition _{}, that is they are assumed to be correlated with correlation
coefficient rÎ(1,1).
For n = 0,1,¼ let L_{n},
be the nth moment with respect to zero of x_{t}, t Î R^{+}, we have:


where p_{*} is the
transition probability density function associated to the stochastic processes x_{t}, v_{t}, t Î R^{+}, implicitly
defined by (2.1), (2.2), (2.3), (2.4). The function p_{*} can be
written as follows:


and the function g_{*}
can be determined proceeding as done in Section 1 to
determine the function g. Note that p_{*} depends on s = tt¢ and not on t and t¢ separately, t > t¢ ³ 0, this
implies that we can rewrite the moments of x_{t}, t Î R^{+}, defined in (2.5) as follows:



(2.8) 
The expressions of the functions D_{j} , j=0,1,2, are (see [7]
for further details):

(2.9) 
_{}
_{}
_{}
Let us choose t¢ = 0, we have x¢ =_{}, v¢ = _{} in (2.9), (2.10), (2.11).
It follows that s = t and that the first three moments of x_{t}, t ÎR^{+}, are given by:
_{}
_{}
_{}
In a similar way using
elementary computations it is possible to obtain the expressions of the moments
L^{*}_{n}(_{}), t Î R^{+},
_{}Î R, _{}ÎR^{+}, for
n > 2 [7]. These expressions become more and more involved
when n increases. Note that formulae (2.13), (2.14) are closed form formulae
containing only elementary functions of quantities that can be observed in the
financial markets. In particular the formulae (2.13), (2.14) do not contain
integrals. These formulae can be used in the formulation of calibration
problems for the
We conclude this section with
an interactive application that given a time value t=t_{1} >0 makes
possible to compare the first order moment L^{*}_{1 }at time
t=t_{1} computed using formula (2.13) with the first order moment
obtained from numerical simulation of the Hull and White model (2.1), (2.2),
(2.3), (2.4). That is given a time value
t=t_{1}>0 the numerically simulated first order moment is obtained
from a sample of the random variable x_{t1} generated
integrating numerically (2.1), (2.2), (2.3), (2.4). The first moment of this sample is computed. In
the interactive application the user can choose the parameters of the