Time iteration

We consider a model with the form:

$\begin{aligned} s_t & = & g\left(m_{t-1}, s_{t-1}, x_{t-1}, m_t \right) \\ 0 & = & E_t \left[ f\left(m_t, s_{t}, x_{t}, m_{t+1}, s_{t+1}, x_{t+1} \right) \right] \end{aligned}$

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 $\varphi$ 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 :
- Taking $\varphi^n$ as the initial guess, find the current period's controls $\varphi^{n+1}(m_t,s_t)$ for any $(m_t,s_t)$ by solving the arbitrage equation : $0 = E_t \left[ f\left(m_t, s_{t}, \varphi^{n+1}(m_t, s_t), g(m_t, s_t, \varphi^{n+1}(m_t, s_t), m_{t+1}), \varphi^{n}(m_{t+1},g(m_t, s_t, \varphi^{n+1}(m_t, s_t), m_{t+1})) \right) \right]$
Repeat until $\eta_{n+1} = \max_{m,s}\left |\varphi^{n+1}(m,s) - \varphi^{n}(m,s) \right|$ is smaller than prespecified criterium $\tau_{η}$

function

dolo.algos.time_iteration.time_iteration(model, dr0=None, verbose=True, details=True, ignore_constraints=False, trace=False, dprocess=None, maxit=1000, inner_maxit=10, tol=1e-06, hook=None, interp_method='cubic', with_complementarities=None)

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

This algorithm iterates on the residuals of the arbitrage equations

Parameters

model (Model) — model to be solved
dr0 (decision rule) — initial guess for the decision rule
verbose (bool) — if True, display iterations
hook (Callable) — function to be called within each iteration, useful for debugging purposes
with_complementarities (bool (True)) — if False, complementarity conditions are ignored

Returns (decision rule :)

approximated solution

dataclass

dolo.algos.results.TimeIterationResult(dr, iterations, complementarities, dprocess, x_converged, x_tol, err, log, trace)

Bases

dolo.algos.results.AlgoResult

TimeIterationResult(dr: object, iterations: int, complementarities: bool, dprocess: object, x_converged: bool, x_tol: float, err: float, log: object, trace: object)