I'm trying to solve a second order difference equation. But there's a stochastic term inside the equation, I was wondering what should the correct way of approaching this problem? Here's the 2nd order equation:
$P_{t}+\lambda\frac{\eta^{s}}{\eta^{d}}P_{t-1}+\frac{(1-\lambda)\eta^{s}}{\eta^{d}}P_{t-1}=\frac{q^{d}-q^{s}}{\eta^d}-\frac{1}{\eta^d}\epsilon_{t}$
Or we could simplify it as $P_{t}+AP_{t-1}+BP_{t-1}=C-D\epsilon_{t}$
where $\epsilon_{t}$ is in iid random variable with zero mean
Should I assume the term is zero or what? Thanks a lot!
Just for your interest, it's derived from the following set of equations by equating (1) with (2), and substitute in (3) eventually. In economics it's called Cobweb model of agricultural production.
System of equations: $Q_{t}^{d}=q^{d}-\eta^dQ_{t}$ ---(1) is the demand equation
$Q_{t}^{s}=q^{s}+\eta^sP_{t}^{d}$ ----(2) is the supply equation
$P_{e}^{t}=\lambda*P_{t-1}+(1-\lambda)E_{t-1}*P_{t}$ ---(3) is how prices adjust
I think I get it. I should probably take expectations on the second order DE equation. The prices in expectations will be substituted by the price adjustment equation.