Solving Dynamic Portfolio Choice Models in Discrete Time Using Spatially Adaptive Sparse Grids
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In this paper, I propose a dynamic programming approach with value function iteration to solve Bellman equations in discrete time using spatially adaptive sparse grids. In doing so, I focus on Bellman equations used in finance, specifically to model dynamic portfolio choice over the life cycle. Since the complexity of the dynamic programming approach—and other approaches—grows exponentially in the dimension of the (continuous) state space, it suffers from the so called curse of dimensionality. Approximation on a spatially adaptive sparse grid can break this curse to some extent. Extending recent approaches proposed in the economics and computer science literature, I employ local linear basis functions to a spatially adaptive sparse grid approximation scheme on the value function. As economists are interested in the optimal choices rather than the value function itself, I discuss how to obtain these optimal choices given a solution to the optimization problem on a sparse grid. I study the numerical properties of the proposed scheme by computing Euler equation errors to an exemplary dynamic portfolio choice model with varying state space dimensionality.
KeywordsSparse Grid Spatial Adaptivity Dynamic Portfolio Choice Value Function Iteration Euler Errors
I thank Julian Valentin for many fruitful discussions on the proposed numerical schemes as well as his support in implementing the MEX file interface to SG++ and parts of the proposed algorithms in MATLAB. I thank Yannick Dillschneider for discussing and helping me to develop the Euler error measure as well for his feedback to draft versions of this paper. I thank two anonymous referees for their rigorous reviews that helped to improve the numerical schemes and the analysis of the results considerably. Valuable feedback was provided by Johannes Brumm, Andreas Hubener, Kenneth Judd, Dirk Pflüger, and Miroslav Stoyanov. I thank the initiative High Performance Computing in Hessen for granting me computing time at the LOEWE-CSC and Lichtenberg Cluster. Finally, I thank Raimond Maurer for supporting this research in every way possible.
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