@ChrisRackauckas the simplest system economists solve is an infinite horizon, 2nd order, boundary value (not initial value), non-linear system of difference equations:
\left[ \begin{array}{c}
(c_{t})^{-\sigma } = \beta (\alpha (k_{t+1})^{\alpha-1} + (1-\delta)) (c_{t+1})^{-\sigma }
\\
k_{t+1} = k_t^\alpha + (1-\delta)k_{t} - c_t
\end{array}\right]
Boundary values:
Initial condition: k(0)=k_0 given
Terminal condition: \lim_{t\to \infty} \beta^t (c_{t})^{-\sigma } k_{t+1} =0
This is easy to solve w/ SolveDSGE.jl. (include the stochastic extensions…)
Can SciML handle this yet?
Does SciML want to eventually?