<p>According to my textbook, the analytical technique for solving a Bellman's Equation is as follows:</p>

<ol>
<li>Guess a form for $V_0(x)$</li>
<li>Solve the maximization problem with respect to the control and obtain a policy function $x' = h_0(x)$</li>
<li>Update the guess by plugging the policy function such that $V_1(x) =F(x,h_0(x)) + \beta V_0(h_0(x))$</li>
<li>Repeat the above until $V_{i+1} = V_i$ At this point, the equation is solved. </li>
</ol>

<p>My question is, why does this technique, specifically the step, $V_{i+1}= V_i$, solve the Bellman equation? 
Thank you!</p>

CC BY-SA 4.0
2018-05-28T23:32:17.250Z
I'm not putting this in an answer because it's too vague but there is probably a proof, that if a function has certain properties ( like the one in 3) and you obtain the fixed point of the solution of that function, then that point is a  maximum. But that's a guess. I would google for "fixed point Bellman" and see what comes up.
mark leeds
CC BY-SA 4.0
2018-05-28T23:38:38.687Z
At a rough glance, it looks the answer is in the paper at the link below  It might be  related to what I said but I would read it carefully as I plan to some day.  https://web.stanford.edu/~pkurlat/teaching/5%20-%20The%20Bellman%20Equation.pdf
mark leeds
<dynamic-programming><bellman-equations>
Explanation of Dynamic Programming "Guess and Verify" Technique
michael_fortunato
# Explanation of Dynamic Programming "Guess and Verify" Technique

(Asked by: username_0 on 2018-05-28)

According to my textbook, the analytical technique for solving a Bellman's Equation is as follows:

1. Guess a form for $V\_0(x)$
2. Solve the maximization problem with respect to the control and obtain a policy function $x' = h\_0(x)$
3. Update the guess by plugging the policy function such that $V\_1(x) =F(x,h\_0(x)) + \beta V\_0(h\_0(x))$
4. Repeat the above until $V\_{i+1} = V\_i$ At this point, the equation is solved.

My question is, why does this technique, specifically the step, $V\_{i+1}= V\_i$, solve the Bellman equation?
Thank you!

**Tags:** dynamic-programming, bellman-equations

> **Comments:**
> **username_1**: I'm not putting this in an answer because it's too vague but there is probably a proof, that if a function has certain properties ( like the one in 3) and you obtain the fixed point of the solution of that function, then that point is a  maximum. But that's a guess. I would google for "fixed point Bellman" and see what comes up.  
> **username_1**: At a rough glance, it looks the answer is in the paper at the link below  It might be  related to what I said but I would read it carefully as I plan to some day.  https://web.stanford.edu/~pkurlat/teaching/5%20-%20The%20Bellman%20Equation.pdf  

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## 0 answers

