Time iteration

We consider an fg model, that is a model with the form:

\(s_t = g\left(s_{t-1}, x_{t-1}, \epsilon_t \right)\)

\(0 = E_t \left[ f\left(s_{t}, x_{t}, s_{t+1}, x_{t+1} \right) \right]\)

where \(g\) is the state transition function, and \(f\) is the arbitrage equation.

The time iteration algorithm consists in approximating the optimal controls, \(x_t = x(s_t)\).

At step \(n\), the current guess for the control, \(x(s_t) = \varphi^n(s_t)\), serves as the control being exercised next period :
  • Given current guess, find the current period’s control by solving the arbitrage equation : \(0 = E_t \left[ f\left(s_{t}, \varphi^{n+1}(s_t), g(s_t, \varphi^{n+1}(s_t)), \varphi^{n}(g(s_t, \varphi^{n+1}(s_t))) \right) \right]\)
dolo.algos.time_iteration.time_iteration(model, initial_guess=None, with_complementarities=True, verbose=True, grid={}, maxit=1000, inner_maxit=10, tol=1e-06, hook=None)

Finds a global solution for model using backward time-iteration.

This algorithm iterates on the residuals of the arbitrage equations

Parameters:

model : NumericModel

“dtmscc” model to be solved

verbose : boolean

if True, display iterations

initial_dr : decision rule

initial guess for the decision rule

with_complementarities : boolean (True)

if False, complementarity conditions are ignored

grid: grid options

maxit: maximum number of iterations

inner_maxit: maximum number of iteration for inner solver

tol: tolerance criterium for successive approximations

Returns:

decision rule :

approximated solution

dolo.algos.time_iteration.residuals_simple(f, g, s, x, dr, dprocess, parms)