# 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, dr0=None, dprocess=None, with_complementarities=True, verbose=True, grid={}, maxit=1000, inner_maxit=10, tol=1e-06, hook=None, details=False)

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

This algorithm iterates on the residuals of the arbitrage equations

Parameters
modelModel

model to be solved

verboseboolean

if True, display iterations

dr0decision rule

initial guess for the decision rule

dprocessDiscretizedProcess (model.exogenous.discretize())

discretized process to be used

with_complementaritiesboolean (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

Returns
decision rule :

approximated solution

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