Dynamic programming and Bellman Equation

We consider the following optimization problem

subject to

where we assume

and

We convert it into the following Bellman equation

at time t

We guess .

Then we get

or

Taking the derivative with respect to c_t, we get

Set

and

for simplicity.

Substituting c together with D1 and D2 back into Bellman equation, we get

since

Comparing the same kinds on both sides,

From the second,

From the first, we get the following temporary result

Here, we can also simplify D₁ and D₂.

We can simplify in the following

So from

we get

Value function would be

From

we also get

Thus