From Assumption 1 the n underlying factors follow a multivariate joint elastic random walk  :
where ± i ( t, x )= a i + B ij x i . In matrix notation this linear system of equations becomes:
The short- term interest rate is expressed as a linear combination of the underlying stochastic factors (Assumption 2), hence:
x - vector of stochastic factors characterising the underlying economic system,
w - vector of weights which are either constants or functions of time.
The solution to (5.1) has the form  :
where ˆ ( t ˆ’ t ) is the matrix solution [ 33 ], [ 3 ]to:
In the special case where B is a constant:
and (5.3) becomes
For this special case  , the expected value and covariance matrix of x ( t ) and x ( t *)( t and t * are future points in time) are then calculated to be  :
where & pound ; is the covariance matrix:
with elements ƒ ij = ij ƒ i ƒ j where ij = [ dz i · dz j ].
 As usual dz i is the standard Wiener process with:
[ dz i ] = 0,
[ dz i dz i ] = dt .
 The deterministic system of equations corresponding to (5.1) is:
This is a linear system, which has a solution of the form
where ½ ( t ) is some function of time and ˆ ( t ˆ’ t ) is the solution to the homogeneous matrix equation
with initial condition ˆ ( t ˆ’ t )= I , and hence it is the fundamental matrix of the system (5.4).
Matrix equation (5.4) now becomes:
and hence the solution to the deterministic system is:
Now consider the system of stochastic equations
which has solution
Applying Ito's Lemma to x ( t )= ˆ ( t ˆ’ t ) ½ ( t )
which is the original differential system (5.1). Hence, we have shown that the solution to this system is
 The calculations are shown for the special case due to the simplified notation, however the more general case proceeds in a similar fashion.
 The covariance is calculated as follows :
and hence (5.7) follows.