I am currently reading this paper on utility maximization in a financial market model. On page 5 the author starts with the case of a finite probability space and on page 19 he considers the incomplete market model, i.e. where the set of equivalent martingale measures is not necessarily a singleton, but under consideration of no arbitrage, i.e. that there exists an equivalent martingale measure. Given a utility function $U: \mathbb{R} \rightarrow \mathbb{R} \cup \{-\infty\}$, on page 13 he defined the Legendre transform of $-U(-\cdot)$ as

$$V(\eta):= \sup_{\xi \in \mathbb{R}} [U(\xi) - \xi \cdot \eta], \text{ for } \eta > 0.$$

Now what confuses me, is that on page 20 in the formula with number (2.74) he uses this conjugate function $V$ for values $$\frac{y \cdot q_n}{p_n},$$ where $p_n$ is assumed to be positive and $q_n:=Q(\omega_n)$ for $Q \in \mathcal{M}^a(S)$, i.e. $Q$ is an absolutely continuous probability measure, such that the assets are martingales under $Q$. In that case we could have that $q_n=0$ for some $n$, since we only have absolute continuity and not equivalence of measures. So that the formula (2.74) I am referring to does not even make sense, since then $$V(\frac{y \cdot q_n}{p_n}) = V(0)$$ and $V$ is not even defined for the value $0$. Now even if we assume we just extend the definition of $V$ to $0$ by setting $$V(0):= \sup_{\xi \in \mathbb{R}}U(\xi),$$ we cannot assure that $V$ is finitely valued, and thus his arguments that the functions $Q \rightarrow \Psi(y,Q)$ is continuous does not even hold.

Are my thoughts wrong, or is there something wrong in this paper?


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