# Second order difference equation with a stochastic term

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

• By the way, can someone advise me how to make the equations appear? I think the latex code is correct right? Thanks! First time using it... – George Apr 25 '15 at 8:53
• What does \eta^P_{t}^{d} mean? $\eta^P_{t}^{d}$ Should it be $\eta^dP_{t}$? – user228113 Apr 25 '15 at 9:04
• Put the LaTeX within dollar signs! – Math1000 Apr 25 '15 at 9:05
• Thanks @Math1000! Yes, you're right @G. Sassatelli – George Apr 25 '15 at 9:14