# Stochastic optimal control problem (calculus)

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$$?$$

• waelde.com/pdf/AIO.pdf you can find the procedure here. Apr 1, 2023 at 20:33
• 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). Apr 1, 2023 at 20:53

## 1 Answer

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.