How do we solve bellman's equation in Merton's model

Question

Studying the expected utility maximization problem in Merton's model, I'm having some difficulties.
Let $t$ be a starting time, $T$ the final finite Time.
We define, \begin{equation} V(t,x)=\underset{\pi \in A}{\sup}\{\mathbb{E}\left[U(X_T(\pi))\right] |X_t=x\} \end{equation} Where $A$ is the set of admissible trading strategies. We can prove that $V$ satisfies (Hamilton-Jacobi-Bellman equation in Merton Model) \begin{equation} \frac{dV(t,x)}{dt} + \sup_{\pi_t} \left( \frac{dV(t,x)}{dx} x (r+ \pi_t(\mu-r)) + \frac{1}{2} \frac{d^2V(t,x)}{dx^2} x^2 \pi_t^2 \sigma^2 \right) = 0 \end{equation}

In the literature (check below for the reference), it's said that $\textit{"a candidate for the optimal control is obtained from the first-order condition}$ $\textit{for the maximum in the HJB equation is :"}$ \begin{equation} \hat{\pi}(t,x)=-\frac{\mu-r}{\sigma^2}\frac{\frac{dV}{dx}}{x\frac{d^2V}{dx^2}}(t,x) \end{equation}

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At this point, I have two questions :
1/ How can we actually obtain / prove this result ?
2/ I don't understand why $V$ is actually present in $\hat{\pi}$ 's expression : By definition, isn't V a sup on all possible $\pi$ ? Shouldn't $\hat{\pi}$ depend on everything apart from $V$ ?
Thanks guys !

Huyen Pham. Optimization methods in portfolio management and option hedging

Answer 1

I'm definitely not an expert on this topic, but it seems to me that:

1. $\hat{\pi}_t$ is here defined as \begin{align} \hat{\pi}_t &= \underset{\pi_t \in \Bbb{R}}{\text{argsup}} \left( \frac{dV(t,x)}{dx} x (r+ \pi_t(\mu-r)) + \frac{1}{2} \frac{d^2V(t,x)}{dx^2} x^2 \pi_t^2 \sigma^2\right) \\ &= \underset{\pi_t \in \Bbb{R}}{\text{argsup}}\, I(\pi_t) \end{align} and is hence obtained by applying the first-order optimality condition: $$ \frac{d I }{d\pi_t}(\hat{\pi_t}) = 0 $$

2. The $\pi$ which appears in the definition of the optimal value function: $$ V(t,x)=\underset{\pi \in A}{\sup}\{\mathbb{E}\left[U(X_T(\pi))\right] |X_t=x\} $$ is a full-blown stochastic process $(\pi_t)_{t \in [0,T]}$. On the other hand, $\pi_t$ which appears in the HJB equation is merely a scalar: it represents the value of the optimal control that applied over the infinitesimal interval $[t,t+dt)$ (hence, "locally", $V(t,x)$ does not depend on it). This is better understood by adopting a dynamic programming view of the problem as mentioned in @Alex C's comment. The idea is then to postulate a candidate optimal control process $\hat{\pi}_t(x)$ by extending the result which has been obtained locally (cf. Bellman's principle of optimality).