From Assumption 1 the n underlying factors follow a multivariate joint elastic random walk ^{ [2] } :
where ± _{ i } ( t, x )= a _{ i } + B _{ ij } x _{ i } . In matrix notation this linear system of equations becomes:
where
The short- term interest rate is expressed as a linear combination of the underlying stochastic factors (Assumption 2), hence:
where
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 ^{ [3] } :
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 ^{ [4] } , 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 ^{ [5] } :
where & pound ; is the covariance matrix:
with elements ƒ _{ ij } = _{ ij } ƒ _{ i } ƒ _{ j } where _{ ij } = [ dz _{ i } · dz _{ j } ].
^{ [2] } As usual dz _{ i } is the standard Wiener process with:
[ dz _{ i } ] = 0,
[ dz _{ i } dz _{ i } ] = dt .
^{ [3] } 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
^{ [4] } The calculations are shown for the special case due to the simplified notation, however the more general case proceeds in a similar fashion.
^{ [5] } The covariance is calculated as follows :
Consider:
and so:
Also:
and hence (5.7) follows.