ABSTRACT
We begin our analysis by supposing that a firm’s investment opportunity set is composed of three interacting variables. These variables might, for example, be the general economic outlook u1(t), the cost of the firm’s primary inputs u2(t) and the demand for the firm’s outputs u3(t), all observed at time t. Now, suppose that the relationship between the instantaneous increments in these variables is defined by the following vector system of stochastic differential equations:⎛⎝ du1(t)du2(t)
du3(t)
⎞⎠= ⎛⎝−2 1 2−4 3 2
−5 1 5
⎞⎠⎛⎝ u1(t)u2(t) u3(t)
⎞⎠dt + ⎛⎝dz1(t)dz2(t)
dz3(t)
⎞⎠ Thus the increment du1(t) = u1(t + dt) − u1(t) in the general economic outlook over the instantaneous period from time t until time t + dt will be
du1(t) = [−2u1(t)+ u2(t)+ 2u3(t)]dt + dz1(t)
where dz1(t) = z1(t + dt) − z1(t) is the increment in the unexpected component of the general economic outlook. Note that dz1(t) has an instantaneous mean Et[dz1(t)] = 0 and an instantaneous variance Vart[dz1(t)] = σ 21 dt (as in §7-7 and §7-8 of Chapter 7). Likewise, the increment in the cost of the firm’s primary inputs over the instantaneous period from time t until time t + dt will be
du2(t) = [−4u1(t)+ 3u2(t)+ 2u3(t)]dt + dz2(t)
where dz2(t) is the increment in the unexpected component of the cost of the firm’s primary inputs and has an instantaneous mean Et[dz2(t)] = 0 and an instantaneous variance Vart[dz2(t)] = σ 22 dt. Finally, the increment in the demand for the firm’s outputs over the instantaneous period from time t until time t + dt will be
du3(t) = [−5u1(t)+ u2(t)+ 5u3(t)]dt + dz3(t)
where dz3(t) is the increment in the unexpected component of the demand for the firm’s outputs and has an instantaneous mean Et[dz3(t)] = 0 and an instantaneous variance Vart[dz3(t)] = σ 23 dt. Moreover, we allow matters to be as simple as possible by assuming that increments in the unexpected components of the three interacting variables – that is, dz1(t),dz2(t) and dz3(t) – are completely uncorrelated. This will mean that the covariances between the unexpected components will all be zero; that is,
Et[dz1(t)dz2(t)] = Et[dz1(t)dz3(t)] = Et[dz2(t)dz3(t)] = 0
where Et(·) is the expectation operator taken at time t (as in §7-3 of chapter 7). We can illustrate how to obtain a solution for the above vector system of stochastic
differential equations by defining
(t) = ⎛⎝ du1(t)du2(t)
du3(t)
⎞⎠
to be the vector containing the increments in the three variables that characterize the firm’s investment opportunity set:
Q = ⎛⎝−2 1 2−4 3 2
−5 1 5
⎞⎠ as the matrix of ‘structural coefficients’,
(t) = ⎛⎝ u1(t)u2(t)
u3(t)
⎞⎠ as the vector containing the levels of the three variables comprising the firm’s investment opportunity set and
d z ∼
(t) = ⎛⎝ dz1(t)dz2(t)
dz3(t)
⎞⎠ as the vector containing the unexpected components of the increments in the variables comprising the firm’s investment opportunity set. With these definitions, we can define the vector system of stochastic differential equations describing the evolution of the variables comprising the firm’s investment opportunity set in the following terms:
(t) = Qu ∼
(t)dt + d z ∼
(t)
Now, if we divide all terms on both sides of this expression by dt, we can restate this vector system of differential equations in the following form:
′(t) = Qu ∼
(t)+ z ∼
′(t)
where
′(t) =
⎛⎜⎜⎜⎜⎜⎜⎝
du1(t)
dt du2(t)
dt du3(t)
dt
⎞⎟⎟⎟⎟⎟⎟⎠ is the vector containing the derivatives of the variables comprising the firm’s investment opportunity set and
′(t) =
⎛⎜⎜⎜⎜⎜⎜⎝
dz1(t)
dt dz2(t)
dt dz3(t)
dt
⎞⎟⎟⎟⎟⎟⎟⎠
is the vector containing the derivatives of the unexpected components of the variables comprising the firm’s investment opportunity set. The components of the vector z
′(t) are often referred to as ‘white noise’ processes; when this is so, dz1(t)
/ dt will be referred to as
a white noise process with variance parameter σ 21 ,dz2(t) / dt will be referred to as a white
noise process with variance parameter σ 22 and dz3(t) / dt will be referred to as a white noise
process with variance parameter σ 23 .
