I see an interesting equation: $\sup _{\pi(t)} E_t\left(d J_t\right)+\lambda V a r_t\left(d\left(e^{r(T-t)} V(t)\right)+d f(t, X(t))\right)=0$,
where,
1.$E_t\left(V^*(T)\right)=e^{r(T-t)} V(t)+E_t\left[\int_t^T \pi^*(s)\left(k(\theta-X(s))+\frac{1}{2} \eta^2+\rho \sigma \eta\right) d s\right]$.
2.$\begin{aligned} J(t, X(t), V(t))= & e^{r(T-t)} V(t)+E_t\left[\int_t^T \pi^*(s)\left\{\left[k(\theta-X(s))+\frac{1}{2} \eta^2+\rho \sigma \eta\right] d s+\eta d W(s)\right\}\right] \\ & +\lambda \operatorname{Var}_t\left[\int_t^T \pi^*(s)\left\{\left[k(\theta-X(s))+\frac{1}{2} \eta^2+\rho \sigma \eta\right] d s+\eta d W(s)\right\}\right] \\ = & e^{r(T-t)} V(t)+c(x, t),\end{aligned}$.
I have the answer is:
$0=\sup _{\pi(t)}\left\{\pi(t)\left[k(\theta-x)+\frac{1}{2} \eta^2+\rho \sigma \eta\right]+\mathbb{D} c+\lambda\left[\eta\left(\pi+f_x\right)\right]^2\right\}$, where:$\mathbb{D} c=c_t+k(\theta-x) c_x+\frac{1}{2} \eta^2 c_{x x}$.
I really have no idea about how to calculate the first and second terms. Could you give me any hint? I will really appreciate it.
I have calculated the third term as:
$d f(t, X(t)))=-\pi^*(s)\left(k(\theta-X(s))+\frac{1}{2} \eta^2+\rho \sigma \eta\right)$.