This is perhaps a trivial question (I’m an economist), but as a non-matematician that might elude me. I have a (two variable) system of nonlinear difference equations (similar to Riccati type differential equations):
$A_{t} =a_{t}+ b_{0}A_{t+1}+b_{1}A_{t+1}B_{t+1}$
$B_{t} =b_{t} + c_{0}B_{t+1}+c_{1}A_{t+1}B_{t+1}$
where $b_{0}\in(0,1)$, $b_{1}\in(0,1)$, $c_{0}\in(0,1)$, $c_{1}\in(0,1)$.
Is it possible to solve for $A_{t}$ (and $B_{t}$) as functions of of current and future $a_{t+i}$ and $b_{t+i}$? For a linear equation, this is trivial (by substituting in future expressions of the difference equations to eliminate $A_{t+i}$ and $B_{t+i}$, given boundedness restrictions so that the limit of $A_{t+i}$ and $B_{t+i}$vanishes), but now the cross terms complicates things. Specifically, is there any know results for systems of this form?