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October 20th, 2017, 11:31 AM   #1
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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 non-stochastic, so I really can't wrap my head around this problem.

Can anyone help me?
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