**1. **

Let us derive a
closed form formula for the transition probability density function of the

Let R and R^{+} be the
sets of real and of positive real numbers respectively and t be a real variable
that denotes time.

Let S_{t}, V_{t}, t Î R^{+}, be real
stochastic processes that describe respectively the asset price and the
associated stochastic variance as a function of time.

The _{t}, V_{t},
t ÎR^{+},
satisfy the following stochastic differential equations (see [10]):

_{}

where r, m, x are real parameters. The
processes W_{t}, Z_{t}, t Î R^{+}, are standard Wiener processes such
that W_{0} = Z_{0} = 0, and dW_{t}, dZ_{t}, t ÎR^{+}, are their
stochastic differentials. Moreover we assume that <dW_{t}^{
}dZ_{t} >= rdt, t>0,
where <· > denotes the expected value of · and rÎ(-1,1) is a
constant known as correlation
coefficient. Equations (1.1) and (1.2) are equipped with the initial
conditions:

_{}

where _{}, _{} are random variables
that we assume to be concentrated in a point with probability one. For
simplicity we identify the random variables _{}, _{} with the points where
they are concentrated. We assume _{}, _{}>
0. The
assumption * *_{}, _{}>
0 with
probability one and (1.1), (1.2) imply that S_{t}, V_{t} > 0 with probability one for t Î R^{+}.

For later convenience we rewrite equations (1.1), (1.2)
using the volatility process v_{t}, t Î R^{+}, instead of the variance process V_{t},
t Î R^{+}.
Recall that we have: V_{t} = v_{t}^{2}, t Î R^{+}. Equations (1.1),
(1.2) become:

_{}

where e = x/2. Note that when r = 0 and m = e^{2} the

Let us introduce the centered log-return x_{t}
= ln(S_{t}e^{-r t}/_{}), t Î R^{+}, and
the quantity _{} =(m/e^{2})-1. Equations (1.5),
(1.6) can be rewritten as follows:

and the
initial conditions (1.3), (1.4) become:

_{}

where _{}, _{} are random variables concentrated in a point with probability one.
Note that_{} is equal to zero with probability one.

Moreover _{}Î R^{+} with
probability one and (1.8) or (1.10) imply that v_{t} > 0 with probability one for t Î R^{+}.

Let _{} be the transition
probability density function of the Hull
and White model (1.7), (1.8), (1.9), (1.10), that is let _{} be the probability density
function of having _{}, _{} when _{}, _{}, t, t’³0, t-t’>0. Note that when t’=0 we must choose
x’=_{}, v’=_{}. In [7] starting from the backward Kolmogorov equation associated
to the stochastic processes defined by (1.7), (1.8) the following formula is
deduced:

_{}

where *i* is the imaginary unit, and the function g is given by:

_{}

where

_{}

_{}

In (1.12) sinh denotes the hyperbolic sine and _{}denotes the Whittaker function of indices a*, *_{}w*.*

When r=0 formula (1.11) reduces to:

_{}