Approximation of discrete dynamic programs with economic applications

Dynamic programming provides a constructive, recursive procedure for computing the optimal decision rule using value function to decentralize stochastic and multiperiod optimization problems into a sequence of simpler deterministic and static optimization problems. A large variety of economic applications including optimal growth, portfolio selection and intertemporal consumption/savings can be formulated in the framework of discrete-time dynamic programs, but they rarely have analytic solutions and thus have to be solved numerically. State variables in economic dynamics such as income or wealth are naturally treated as continuous quantities, whereas solutions to continuous-state dynamic programs are generally approximated by discretization or smooth approximation methods.; This thesis first focuses on the numerical analysis of dynamic programming with discrete states. Convergence properties and error bounds are given for new versions of Jacobi and Gauss-Seidel value iterations. Upwind Gauss-Seidel iteration, a new algorithm featuring state reordering for convergence acceleration, is discussed and its superiority over other value iterations for economic dynamics with absorbing states is shown by numerical examples.; For economic problems with the present value criterion, most smooth interpolation methods approximating the value functions do not preserve monotonicity and concavity properties which inherit from the utility functions in the optimization objectives. We propose shape-preserving methods to better approximate the solutions of continuous-state dynamic programs. A unified approach on the error analyses of quadratic spline and cubic Hermite interpolation is presented, and the choice of grid size to make the latter method nape preserving is also elaborated. We finally apply bivariate interpolation with Bernstein polynomials to solve a numerical example on pension savings allocation. This utilization of shape-preserving methods to approximate the solutions of two-dimensional dynamic programming problems is unprecedented.