Finite differences on sparse grids for continuous time heterogeneous agent models

Abstract

We present a finite difference method working on sparse grids to solve higher dimensional heterogeneous agent models. If one wants to solve the arising Hamilton-Jacobi-Bellman equation on a standard full grid, one faces the problem that the number of grid points grows exponentially with the number of dimensions. Discretizations on sparse grids only involve <em>O</em>(<em>N</em>(log<em>N</em>)^{<em>d</em>−1}) degrees of freedom in comparison to the <em>O</em>(<em>N^d</em>) degrees of freedom of conventional methods, where <em>N</em> denotes the number of grid points in one coordinate direction and <em>d</em> is the dimension of the problem. Whereas one can show convergence for the used finite difference method on full grids by using the theory introduced by Barles and Souganidis [4], we explain why one cannot simply use their results for sparse grids. Our numerical studies show that our method converges to the full grid solution for a two-dimensional model. We analyze the convergence behavior for higher dimensional models and experiment with different sparse grid adaptivity types.