ABSTRACT
We can apply similar procedures to the differential equation describing the evolution of the variable y2(t), in which case it follows that
y2(t) = c2e2t + t∫
e2(t−s)[dz2(s)− dz1(s)]
where, as before, c2 is a constant of integration. Likewise, from the differential equation for the variable y3(t), we have
y3(t) = c3e3t + t∫
e3(t−s)[dz3(s)− dz1(s)]
where, again, c3 is a constant of integration. Moreover, having obtained solutions for each of the transformed variables, we are now in a position to determine the solution to our original vector system of differential equations, namely,
(t) = My ∼
(t)
or ⎛⎝ u1(t)u2(t) u3(t)
⎞⎠= ⎛⎝ 1 1 11 2 1
1 1 2
⎞⎠ ⎛⎜⎜⎝
c1et + ∫ t 0 e
t−s[3dz1(s)− dz2(s)− dz3(s)] c2e2t +
∫ t 0 e
2(t−s)[dz2(s)− dz1(s)] c3e3t +
∫ t 0 e
3(t−s)[dz3(s)− dz1(s)]
⎞⎟⎟⎠ Evaluating this transformation equation will therefore show that the variable describing the general economic outlook, u1(t), will evolve in accordance with the following process:
u1(t) = c1et + c2e2t + c3e3t
+ t∫
(3e(t−s) − e2(t−s) − e3(t−s))dz1(s)+ t∫
(e2(t−s) − e(t−s))dz2(s)
+ t∫
(e3(t−s) − e(t−s))dz3(s)
Likewise, the variable describing the cost of the firm’s primary inputs, u2(t), will evolve in accordance with the following process:
u2(t) = c1et + 2c2e2t + c3e3t
+ t∫
(3e(t−s) − 2e2(t−s) − e3(t−s))dz1(s)+ t∫
(2e2(t−s) − e(t−s))dz2(s)
+ t∫
(e3(t−s) − e(t−s))dz3(s)
Finally, the variable describing the demand for the firm’s outputs, u3(t), will evolve in accordance with the following process:
u3(t) = c1et + c2e2t + 2c3e3t
+ t∫
(3e(t−s) − e2(t−s) − 2e3(t−s))dz1(s)+ t∫
(e2(t−s) − e(t−s))dz2(s)
+ t∫
(2e3(t−s) − e(t−s))dz3(s)
Note also that if we set t = 0 in the above transformation matrix it follows that
(0) = ⎛⎜⎝ u1(0)u2(0)
u3(0)
⎞⎟⎠= ⎛⎝ 1 1 11 2 1
1 1 2
⎞⎠⎛⎝ c1c2 c3
⎞⎠= My ∼
(0)
or ⎛⎝ c1c2 c3
⎞⎠= ⎛⎝ 1 1 11 2 1
1 1 2
⎞⎠−1⎛⎝ u1(0)u2(0) u3(0)
⎞⎠= ⎛⎝ 3 −1 −1−1 1 0
−1 0 1
⎞⎠⎛⎝u1(0)u2(0) u3(0)
⎞⎠= M−1u ∼
(0)
This in turn will mean that the integration constants in the above expression for the variables comprising the firm’s investment opportunity set will turn out to be
c1 = 3u1(0)− u2(0)− u3(0),c2 = u2(0)− u1(0) and c3 = u3(0)− u1(0)
We can use these results to determine the distributional properties of the variables comprising the firm’s investment opportunity set. Thus, if we use the expressions for the integrating constants given here, it follows that for the first variable, the general economic outlook u1(t), we have
u1(t) = [3u1(0)− u2(0)− u3(0)]et +[u2(0)− u1(0)]e2t +[u3(0)− u1(0)]e3t
+ t∫
(3e(t−s) − e2(t−s) − e3(t−s))dz1(s)+ t∫
(e2(t−s) − e(t−s))dz2(s)
+ t∫
(e3(t−s) − e(t−s))dz3(s)
We can take expectations across this expression and thereby show that the variable describing the general economic outlook will be normally distributed with a mean
E0[u1(t)] = [3u1(0)− u2(0)− u3(0)]et +[u2(0)− u1(0)]e2t +[u3(0)− u1(0)]e3t
Moreover, we can apply Wiener’s Theorem (as in §7-5 of Chapter 7) in conjunction with the fact that dz1(t),dz2(t) and dz3(t) are mutually uncorrelated (as in §8-2 above) to show that the variance of the general economic outlook variable will be
Var0[u1(t)] =
(3e(t−s) − e2(t−s) − e3(t−s))2ds+σ 22 t∫
(e2(t−s) − e(t−s))2ds
(e3(t−s) − e(t−s))2ds
Similar calculations can be applied to determine the distributional properties of the other two variables, u2(t) and u3(t).
