Let’s consider the Forward-Backward SDE composed of one Forward SDE:
dx = a(x, y) * dt + b(x, y) * dW
and one Backward SDE:
dy = c(x, y) * dt + z * dW
where a, b and c are some functionals. Thanks to Dynare, I can solve this FBSDE using a second order approximation near its steady state using
dt = 1/12; model; x - x(-1) = a(x(-1), y) * dt + b(x(-1), y) * dWt; y(+1) - y = c(x(-1), y) * dt; end; shocks; var dWt = dt; end;
Let’s assume now the more general case where a, b and c depend on z. I would need to define z as an endogenous variable, but what equation should be associated with it?
In other words, how can I access the volatility of an endogenous forward-looking variable?
Thanks a lot for your help!