Boundary Values of the Utility Conjugate in Incomplete Markets
Summary
The document examines a potential domain issue in a paper on utility maximization in finite probability spaces. It describes a conjugate function defined for strictly positive arguments, then questions a later expression that evaluates the function at a value proportional to a state probability under an absolutely continuous martingale measure. Absolute continuity permits that probability to be zero, which would place the argument outside the stated domain.
The author considers whether extending the conjugate to zero by taking the supremum of the utility function resolves the issue. They note that this supremum may be infinite, potentially challenging a continuity claim used in the paper’s argument. The text provides no resolution or independent verification of the cited paper’s formula, so the alleged flaw remains a question rather than an established error. Its value is in highlighting a boundary-domain condition that readers may need to check when applying duality arguments.
Key ideas
- Absolute continuity allows a martingale measure to assign zero probability to some states.
- A conjugate defined only for positive arguments may therefore be evaluated outside its stated domain.
- Extending its definition to zero can produce an infinite value, depending on the utility function.
- The document raises a possible impact on a continuity argument but does not resolve or verify the issue.
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Full text
# Utility Maximization on a finite Probability Space. Possible mistakes in a paper?
# Utility Maximization on a finite Probability Space. Possible mistakes in a paper?
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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.