By Karl Hinderer, Ulrich Rieder, Michael Stieglitz
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Extra info for Dynamic Optimization: Deterministic and Stochastic Models (Universitext)
1 ˇ/]. 1(c3). (b4) Let u be concave. s/ Ä s0 s 0 for s Ä s . x/ D zx, and u1 D u. (b5) Let u be convex and ˇ < 1. s0 / is s0 optimal for all N m. 5. (b6) The value functions Vn , n 1, are Lipschitz continuous in d and also in ˇ, both uniformly in s. 1(c6). 2), in case ˇ < 1 the sequence of value functions converges for n ! 1 uniformly to some function V. 2, and V cannot be defined recursively. s/ for some d0 2 RC , ˇ 2 Œ0; 1. s/ if and only if 0n 1 at Ä s. s0 /. s0 /. (d) if u is increasing, then s 7!
S/ is a finite subset of R then there exists for n 1 a smallest maximizer at stage n. (b) All value functions are upper bounded (hence also finite) if r and V0 are upper bounded. All value functions are bounded if r and V0 are bounded. (c) All minimal cost functions are lower bounded (hence also finite) if c and C0 are lower bounded, in particular if c 0 and C0 0. All minimal cost functions are bounded if c and C0 are bounded. 3(c). s/ 2 R. The second assertion is obviously true. ˇ/ sup jrj C ˇ sup jV0 j.
S/ D u. s//, n 1, where V0f WD V0 . Now one easily obtains, using induction on n 0, that kV Vnf k Ä 30 2 The Stationary Deterministic Model and the Basic Solution Procedure ˇ n kV V0 k. s/ when n tends to 1. Now we turn to continuous DPs. 1 below where only a discrete model makes sense. 7 below. Here are a few comments on the appropriateness of discrete or continuous versions and on their solutions. (i) From a rigorous point of view continuous DPs cannot be completely realistic models for applications since they assume infinite divisibility of states and/or of actions.