# Stochastic difference equation?

## Homework Statement

This is a question about one single step of a solution of a long equation.

where P, U and V are variables. a, b, c, d are constants and t is the time, which are measured in discrete periods.

The question is how to go from equation 1 to equation 3, and how the L appears.

Attempted solution

I have solved it until equation 2, and I see that the solution requires the extraction of P out of the bracket on the left hand side. Problem becomes how to do it, since the P minus one period doesn't equal P of current period. Nevertheless the key somehow does this. Does this require some other method than simple algebra?

The key mentions it is a "stochastic difference equation".

Just looking at it, L appears to be a shift operator, that is $Lp_t=p_{t-1}$. Note that $Lp_t$ is not multiplication, but rather the L operator is applied to $t_p$. Another example of this kind of notation involves the differential operator $D_x$, in particular $D_xf(x)$ is the derivative of f(x) w.r.t. x.