
Applied Math Applied Math Forum 
 LinkBack  Thread Tools  Display Modes 
October 20th, 2017, 11:31 AM  #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? 

Tags 
bellman, contraction, equation, mapping 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Banach's Contraction Mapping Theorem  zactops  Topology  2  December 3rd, 2016 07:29 AM 
Applying the Contraction Mapping Principle  guynamedluis  Real Analysis  1  November 19th, 2011 12:23 PM 
Contraction Mapping and Diagonally Dominant Jacobian  needmath  Applied Math  0  August 14th, 2011 03:36 PM 
Contraction Mapping  six  Real Analysis  1  October 31st, 2010 07:19 PM 
contraction mapping theorem  aptx4869  Real Analysis  1  April 20th, 2007 10:10 PM 