Original article (link) posted: 28/09/2005
In the last class, we saw the following results;
Necessity: the value function of the dynamic problem V becomes the fixed point of the corresponding Bellman problem, i.e., TV=V.
Sufficiency: if the bounded function U satisfies TU=U in a Bellman problem and if there exists a maximizing selection for TU, then U is a value function of the original dynamic problem.
Our focus on Lecture 3 is to prove these results formally and to derive some other properties about Bellman operator T.
Topics in the class
1) The proof of necessity
2) The proof of sufficiency
To read the chapter 4 of SLP in advance helped me a lot to understand the proof. Although Chapter 4 deals with DP under certainty, it has direct link to uncertain case which was covered in the class.
3) Finding a value function
(a) The Bellman operator is a contraction (by Blackwell)
(b) The set of bounded and continuous functions with sup-norm is a complete metric space. Completeness is quite important to derive properties of value functions, because it assures the properties in the limit (=the value function).
Blackwell's sufficient conditions for a mapping to be a contraction seem very useful. Professor Dutta mentioned that despite the seminal works in the field of applied mathematics Blackwell's first half of the career was not bright because of the racial discrimination. (He is a black.)