Time iteration

We consider a model with the form:

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

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

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

The time iteration algorithm consists in approximating the optimal controls as a function of exogenous and endogenous controls \(x_t = \varphi(m_t,s_t)\).

At step \(n\), the current guess for the control, \(x(s_t) = \varphi^n(m_t, s_t)\), serves as the control being exercised next period :

  • Given current guess, find the current period’s \(\varphi^{n+1}(m_t,s_t)\) controls for any \((m_t,s_t)\) by solving the arbitrage equation : \(0 = E_t \left[ f\left(m_t, s_{t}, \varphi^{n+1}(m_t, s_t), g(s_t, \varphi^{n+1}(m_t, s_t)), \varphi^{n}(m_{t+1},g(s_t, \varphi^{n+1}(s_t))) \right) \right]\)
dolo.algos.time_iteration.time_iteration(model, initial_guess=None, dprocess=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

model : Model

model to be solved

verbose : boolean

if True, display iterations

initial_guess : decision rule

initial guess for the decision rule

dprocess : DiscretizedProcess (model.exogenous.discretize())

discretized process to be used

with_complementarities : boolean (True)

if False, complementarity conditions are ignored

grid: grid options

overload the values set in options:grid section

maxit: maximum number of iterations
inner_maxit: maximum number of iteration for inner solver
tol: tolerance criterium for successive approximations
hook: Callable

function to be called within each iteration, useful for debugging purposes

decision rule :

approximated solution

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