<p>I think there should be a minus before the final term in the expression you are looking for. In any case, you did not plug in the optimality condition for <span class="math-container">$w$</span>, which is in your case</p>
<p><span class="math-container">$$
w = -\frac{V'(A)}{AV''(A)}\frac{\mu - r}{\sigma^2}.
$$</span>
Plug that in for <span class="math-container">$W$</span> in the final equation in your post to get the desired expression.</p>

CC BY-SA 4.0
2023-04-02T13:19:53.010Z
2023-04-02T13:19:53.010Z
Wittgenstein's Poker
<p>I am reading a paper with a stochastic optimal control problem. At one point the author faces the following Hamilton-Jacobi-Bellman (HJB) equation:</p>
<p><span class="math-container">$$\rho V(A)=\max_{c,w}\left\{ u(c)+V'(A)\left[(r(1-w)+\mu w)A+y-c\right]+\frac{1}{2}w^{2}\sigma^{2}A^{2}V''(A)\right\}$$</span></p>
<p>He writes that: <em>If <span class="math-container">$u''(\cdot)&lt;0$</span> then the optimal policy function for <span class="math-container">$c$</span> may be written as: <span class="math-container">$c^{*}=C(A)=(u')^{-1}(V'(A))$</span>. Substituting into the HJB equation, we get the differential equation over <span class="math-container">$V(A)$</span></em>:</p>
<p><span class="math-container">$$\rho V(A)= u(C(A))+V'(A)(y+rA-C(A))+\frac{1}{2}\left(\frac{r-\mu}{\sigma}\right)^{2}\frac{(V'(A))^{2}}{V''(A)}$$</span></p>
<p>I can easily see the replacement in the first part of the latter equation but do not understand how the last term <span class="math-container">$\frac{1}{2}\left(\frac{r-\mu}{\sigma}\right)^{2}\frac{(V'(A))^{2}}{V''(A)}$</span> is obtained. If i try to do some algebra the most i can get is:</p>
<p><span class="math-container">$$\rho V(A)= u(C(A))+V'(A)(y+rA-C(A))+WA\left[(\mu-r)V'(A)+\frac{1}{2}WA\sigma^{2}V''(A)\right]$$</span></p>
<p>Is there anyone who can help me understand what the correct procedure is to obtain the result of the paper<span class="math-container">$?$</span></p>

CC BY-SA 4.0
2023-04-01T20:33:25.003Z
http://waelde.com/pdf/AIO.pdf you can find the procedure here.
optimal control
CC BY-SA 4.0
2023-04-01T20:53:44.990Z
I do not know exactly but you can probably find in the stochastic continuous part. The book has four sections deterministic models (discrete/continuous) and stochastic models (discrete/continuous).
optimal control
<mathematical-economics><stochastic-processes><linear-algebra><stochastic-calculus>
Stochastic optimal control problem (calculus)
Alessandro
# Stochastic optimal control problem (calculus)

(Asked by: username_0 on 2023-04-01)

I am reading a paper with a stochastic optimal control problem. At one point the author faces the following Hamilton-Jacobi-Bellman (HJB) equation:

$$\rho V(A)=\max\_{c,w}\left\{ u(c)+V'(A)\left[(r(1-w)+\mu w)A+y-c\right]+\frac{1}{2}w^{2}\sigma^{2}A^{2}V''(A)\right\}$$

He writes that: *If $u''(\cdot)<0$ then the optimal policy function for $c$ may be written as: $c^{\*}=C(A)=(u')^{-1}(V'(A))$. Substituting into the HJB equation, we get the differential equation over $V(A)$*:

$$\rho V(A)= u(C(A))+V'(A)(y+rA-C(A))+\frac{1}{2}\left(\frac{r-\mu}{\sigma}\right)^{2}\frac{(V'(A))^{2}}{V''(A)}$$

I can easily see the replacement in the first part of the latter equation but do not understand how the last term $\frac{1}{2}\left(\frac{r-\mu}{\sigma}\right)^{2}\frac{(V'(A))^{2}}{V''(A)}$ is obtained. If i try to do some algebra the most i can get is:

$$\rho V(A)= u(C(A))+V'(A)(y+rA-C(A))+WA\left[(\mu-r)V'(A)+\frac{1}{2}WA\sigma^{2}V''(A)\right]$$

Is there anyone who can help me understand what the correct procedure is to obtain the result of the paper$?$

**Tags:** mathematical-economics, stochastic-processes, linear-algebra, stochastic-calculus
> **Comments:**
> **username_1**: http://waelde.com/pdf/AIO.pdf you can find the procedure here.  
> **username_1**: I do not know exactly but you can probably find in the stochastic continuous part. The book has four sections deterministic models (discrete/continuous) and stochastic models (discrete/continuous).  

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## 1 answers

### Answer by username_2 on 2023-04-02. Score: 1

I think there should be a minus before the final term in the expression you are looking for. In any case, you did not plug in the optimality condition for $w$, which is in your case

$$
w = -\frac{V'(A)}{AV''(A)}\frac{\mu - r}{\sigma^2}.
$$
Plug that in for $W$ in the final equation in your post to get the desired expression.


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