Note: this is a preliminary and incomplete: do not quote without permission of authors. 3 In this vein section 5 discusses the desirability of solving a slightly perturbed version of Bellman's equation with an approximate Bellman operator r, that is everywhere differentiable (unlike r which has kinks at certain points V E B). 4 In some cases parametric methods allow us to exploit certain types of prior information we might have about the solution V, i.e. monotonicity, convexity, etc. For

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... , see Judd, 1994. 5 Consistency of parametric methods also depends on being able to find arbitrarily accurate solutions to the associated nonlinear optimization and multivariate integration problems. 6 There is a closely literature on discretization and parametric approximation of abstract operator equations (Cryer, 1982 , KrasnosePsIdi et. al. 1982, and Rall, 1982 which can be viewed as encompassing the methods outlined in this chapter as a special case. However this literature and the closely related literature on approximation of collectively compact operators (Anselone, 1971 and Anselone and Ansorge, 1981) is rather abstract and it is often easier to prove the convergence of a particular method from first principles than to verify the general sufficient conditions required for these more general consistency results.