Dynamic programming and Bellman Equation

We consider the following special Bellman equation with Epstein-Zin preference

at time t

where

and

where is random return that is strictly positive.

Basically, we use the following two:

(1) The part is a linear function of st.

(2) Envelope condition with respect to Wt

to simplify the problem.

We guess

Then Bellman equation is going to be

where

FOC w.r.t. st is

where

by function of function rule many times.

Firstly, from envelope condition with respect to Wt, we have

Thus

It means that the first part factored out is some positive when.

Then we can say that

since the part factored out is positive. That is

Secondly, we set and where B is some constant because

or

where it is a linear function of st and we can set

for some constant B.

Now

is set to be

using this constant B.

where let us define it as G for simplicity.

So

Plugging it into Bellman equation, we get

Comparing the coefficients, we finally get

Since both B and G are functions of A, all we have to do is solve this for A.

We have solved out and the guess is right.