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4Quadrants

4 Quadrants of Dynamic Optimization

$V(s_{t}) \text{ } \equiv \text{ } \underset{ c_{t} }{ \sup } \text{ } E_{t}\left[ \int_{s=t}^{s=T} e^{-\rho (s-t)} u(c_t, s_t) dt \right] $
$ds_{t} = \mu(s_{t},c_{t})dt + \sigma(s_{t},c_{t})dZ_{t} $
$c(s_{t}) \text{ } \equiv \text{ } \underset{ c_{t} }{ \arg\sup } \text{ } E_{t}\left[ \int_{s=t}^{s=T} e^{-\rho (s-t)} u(c_t, s_t) dt \right] $

1: Discrete-time/deterministic solution using SolveDSGE.jl (@RJDennis). Code link. The corresponding simulations (using 6 different methods):

image

2: Discrete-time/deterministic solution using POMDPs.jl (@zsunberg) -Code link. See discussion here. image image

3: Discrete-time/deterministic solution using JuMP.jl (@odow & co) -Please post the code below.

4: Discrete Time/Stochastic, Difference Equations SolveDSGE.jl (@RJDennis) -Code link. The corresponding simulations (using perturbation methods):

image

5: Discrete Time/Stochastic, Bellman Equations POMDPs.jl (@zsunberg)To Do. -Please post code below using e.g. a 2-state Markov Chain for z.

6: Continuous Time/Deterministic, Bellman Equations (my HJB Solver, based on Ben Moll) -Code link. The corresponding simulations:

image

7: Continuous Time/Deterministic, Bellman Equations EconPDEs.jl (@matthieu) To Do...

8: Continuous Time/Deterministic, Hamiltonian InfiniteOpt.jl (@pulsipher) -To Do. Only finite horizon currently.

9: Continuous Time/Deterministic, DE SciML.jl (@ChrisRackauckas). -TO DO. Only finite horizon currently. If someone knows how to transform the problem to solve an infinite horizon system please post code below.

10: Continuous Time/Stochastic, Bellman Equations @matthieu (Custom HJB Solver & EconPDEs.jl). To Do.

To Do: Deep learning methods https://discourse.julialang.org/t/solving-hjb-pde-using-deep-learning/60639/13

Transform finite horizon DE, into infinite Horizon DE: https://discourse.julialang.org/t/solving-boundary-value-differential-equation-problems-for-economics/72871

TO Do: incorporate discrete time (deterministic/stochastic) w/ VFI https://github.com/shanemcmiken/PEinvestment Except try to make generic