Finite‐Difference Methods for Continuous‐Time Dynamic Programming

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Abstract
Introduces some of the methods and underlying ideas behind computational fluid dynamics—in particular, the use is discussed of finite‐difference methods for the simulation of dynamic economies. A standard stochastic dynamic programming model is considered of a macroeconomy. Finite‐difference methods are applied to this problem (model), resulting in a second‐order nonlinear partial differential equation that has some features in common with the governing equations of fluid dynamics; the idea is also introduced of ‘upwind’ or solution‐dependent differencing methods, and the stability of these is discussed through the analysis of model problems. An implicit solution to the nonlinear dynamic programming problem is then developed and tested, with the motivation of reducing the computer time required to solve it. Finally, the extension of the finite‐difference method to a two‐state dynamic programming problem is considered.