When Is the Optimal Wealth Martingale Integrand a Martingale?

Summary

The document outlines the martingale method for expected-utility maximization in a complete-market style setup. It describes replacing optimization over trading strategies with a static choice of terminal wealth subject to a budget constraint under a pricing measure. A Lagrange multiplier then yields optimal terminal wealth as a function of the Radon–Nikodym density. The wealth process can be represented through a stochastic integral under the pricing measure, whose predictable integrand corresponds to the control or strategy exposure.

The central issue raised is whether that optimal integrand is itself always a martingale under the pricing measure. The author reports seeing examples where it appears to equal the conditional expectation of its terminal value, and asks whether this property can be proved generally. No proof, counterexample, assumptions, or resolution is supplied, so the note identifies a theoretical question rather than establishing a general result. The martingale property of wealth alone does not, from the discussion given, settle the property of its integrand.

Key ideas

• The martingale method converts dynamic utility maximization into a constrained choice of terminal wealth.\nThe optimal terminal wealth is characterized as a function of the pricing measure density.\nThe wealth process is represented as a martingale stochastic integral with a predictable integrand.\nThe document asks whether this integrand is generally a martingale, but gives no answer.

Tags

# Martingale method for utility maximization - is the optimal strategy also a martingale?


# Martingale method for utility maximization - is the optimal strategy also a martingale?












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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