The Martingale Method for utility maximization (seen in e.g. Björk's book) is based on separating the optimization problem $E^\mathbb{P}[U(X_T)]$ over a class of admissible strategies into the static problem of determining the optimal wealth profile $\hat{X}_T$. If the inital capital is $x$ the constraint is $E^\mathbb{Q}[X_T]=x$ hence the problem turns into maximizing the Lagrangian

\begin{equation} E^\mathbb{P}[U(X_T) - \lambda(L_T X_T - x)] \end{equation}
where $L_T$ is the Radon–Nikodym derivative and $\lambda$ is the lagrange multiplier. Solving this gives the optimal wealth as some function of $L_T$, $X_T = X_T(L_T)$, and since the martingale dynamics is $d\hat{X}_t = \hat{u}_t dW_t^\mathbb{Q}$ (for some predictable $\hat{u}_t$) all that is left is to determine $\hat{u}_t$ (e.g. by Ito's lemma).

Now to my question: I have noticed that for the problems I have encountered $\hat{u}_t$ is also a $\mathbb{Q}$-martingale. For example it might be easy to determine $\hat{u}_T$ close to the terminal date $T$, and then it is evident that $\hat{u}_t = E^\mathbb{Q}[\hat{u}_T | \mathcal{F}_t]$ after the problem has been solved. Is it possible to prove that this is true? If $\hat{u}_t$ is known this is obviously easy to check, but what can be said about the general case?


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