By Karl Hinderer, Ulrich Rieder, Michael Stieglitz
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Example text
S; /. 8). 0; 2; 7; 15; 27; 44; 67; 97; 135; 182; 239/. 7; see also Fig. 1. 10/ is 10- optimal. x/ D 0g. s/ is s-optimal. This was due to the fact that the system which starts at some state s0 2 J0 stays there and earns neither rewards nor a terminal reward. There are many DPs with the same property, which is formalized as follows. 40 3 Additional General Issues 250 C10 (·) discrete C10 (·) continuous 200 150 100 50 0 0 1 2 3 4 5 s 6 7 8 9 10 Fig. s/. s/. s/ D 0 for all s 2 J0 . The set J WD S J0 is called the essential state space (with respect to J0 ).
Sa0 /. Then DP0 has the value functions, starting with V00 WD du, ˚ s 7! 13) which intuitively equals Vn . s; a0 /. s/ is a maximizer at stage n in DP. 14) for n 1. g. a/ D a (cf. 6(a)) or if ˇz D 1 (cf. 1). However, as we now indicate, for relatively general utility u the subsequent structural properties (b1)–(b7) of the solution are valid. s/, n 2 The Stationary Deterministic Model and the Basic Solution Procedure 1, is increasing in s, non-negative and finite. s/ D 0. s/ is s-optimal. s/ D 0 by continuity of u.
40 3 Additional General Issues 250 C10 (·) discrete C10 (·) continuous 200 150 100 50 0 0 1 2 3 4 5 s 6 7 8 9 10 Fig. s/. s/. s/ D 0 for all s 2 J0 . The set J WD S J0 is called the essential state space (with respect to J0 ). We do not call ; and S absorbing (although they trivially have the property of an absorbing set) since these two cases do not contain useful information. 2 (The value function on an absorbing set) In a DP with absorbing set J0 we have Vn D 0 on J0 for all n 0: Thus one must perform the VI only for s in the essential state space.