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October 20th, 2017, 12:31 PM  #1 
Newbie Joined: Oct 2017 From: Italy Posts: 2 Thanks: 0  Bellman Equation and Contraction Mapping
Hello all. I'm taking a grad course where optimization problems are part of the syllabus, and I'm afraid I'm not really meeting all math requirements to take the course but alas, I had no choice but to take it. I have this problem (click for link) and I'm kinda stuck. Now, I know that to prove $\displaystyle T$ is a contraction mapping I should first prove the monotonicity, and then the discounting. Below you'll find how I did it. Monotonicity: $\displaystyle f(s) \leq g(s)$ $\displaystyle \sup[\pi(s)x+\beta \int f(s')p(s's,x)] \leq \sup[\pi(s)x+\beta \int g(s')p(s's,x)]$ $\displaystyle (Tf)(s) \leq (Tg)(s)$ Discounting: $\displaystyle T(f+c)(s) \leq Tf+\beta c$ $\displaystyle T(f+c)(s)=\sup[\pi(s)x+\beta \int f(s')p(s''s,x)+c]=Tf(s)+\beta c$ Is this enough, and is it correct? Or am I missing something? The only example I have in my notes is nonstochastic, so I really can't wrap my head around this problem. Can anyone help me? 

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bellman, contraction, equation, mapping 
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