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Nonconcave Robust Utility Maximization under Projective Determinacy

Laurence Carassus, Massinissa Ferhoune

TL;DR

The paper tackles robust utility maximization under Knightian uncertainty in a discrete-time market, where utilities and market objects are allowed to be nonconcave and random. The authors introduce a projective-measurability framework and rely on the Projective Determinacy (PD) axiom to guarantee the existence of an optimal investment strategy in multi-period settings, provided the utility is upper-semicontinuous; they also establish a one-step strategy with explicit bounds and extend optimality to a broad class of Type (A) utilities under suitable integrability. A key contribution is the dynamic programming construction that remains well-defined in the projective setting, enabling measurable selections and a glued strategy that achieves near-optimality, with exact optimality recovered under usc and non-discontinuity conditions. The paper also clarifies the necessity of PD by presenting counterexamples and discusses implications for non-ZFC frameworks, connecting to quasi-sure no-arbitrage and robust market models such as nondominated Black–Scholes and binomial markets.

Abstract

We study a general robust utility maximization problem in a discrete-time frictionless market. The investor is assumed to have a possibly infinite, random, nonconcave, and nondecreasing utility function defined on the whole real line. She also faces model ambiguity on her beliefs about the market, which is modelled through a set of priors. We assume that the utility and the prices are projective functions of the path, while the graphs of the local priors are projective sets. Our other assumptions are stated on a prior-by-prior basis and correspond to generally accepted assumptions in the literature on markets without ambiguity. Under the set-theoretic axiom of Projective Determinacy (PD), our main result is the existence of an optimal investment strategy when the utility function is also upper-semicontinuous. We further provide several counterexamples justifying our assumptions.

Nonconcave Robust Utility Maximization under Projective Determinacy

TL;DR

The paper tackles robust utility maximization under Knightian uncertainty in a discrete-time market, where utilities and market objects are allowed to be nonconcave and random. The authors introduce a projective-measurability framework and rely on the Projective Determinacy (PD) axiom to guarantee the existence of an optimal investment strategy in multi-period settings, provided the utility is upper-semicontinuous; they also establish a one-step strategy with explicit bounds and extend optimality to a broad class of Type (A) utilities under suitable integrability. A key contribution is the dynamic programming construction that remains well-defined in the projective setting, enabling measurable selections and a glued strategy that achieves near-optimality, with exact optimality recovered under usc and non-discontinuity conditions. The paper also clarifies the necessity of PD by presenting counterexamples and discusses implications for non-ZFC frameworks, connecting to quasi-sure no-arbitrage and robust market models such as nondominated Black–Scholes and binomial markets.

Abstract

We study a general robust utility maximization problem in a discrete-time frictionless market. The investor is assumed to have a possibly infinite, random, nonconcave, and nondecreasing utility function defined on the whole real line. She also faces model ambiguity on her beliefs about the market, which is modelled through a set of priors. We assume that the utility and the prices are projective functions of the path, while the graphs of the local priors are projective sets. Our other assumptions are stated on a prior-by-prior basis and correspond to generally accepted assumptions in the literature on markets without ambiguity. Under the set-theoretic axiom of Projective Determinacy (PD), our main result is the existence of an optimal investment strategy when the utility function is also upper-semicontinuous. We further provide several counterexamples justifying our assumptions.
Paper Structure (20 sections, 24 theorems, 184 equations)

This paper contains 20 sections, 24 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Setting and main result
  3. Projective setup
  4. Financial Setting
  5. Direct Assumptions on $U$
  6. Value functions and Assumption on $U_0^P$
  7. Main result
  8. Application
  9. One period case
  10. Multiple-period case
  11. Proof of Theorem \ref{['one_step_strategy_pj']}
  12. Proof of Theorem \ref{['optimality_M_typeA_pj']}
  13. Counterexamples
  14. $U$ need to be usc
  15. If $V$ is continuous, $\Psi$ may be fail to be usc
  16. ...and 5 more sections

Key Result

Theorem 1

Assume the (PD) axiom. (i) Let $X$ be a Polish space, then $\textbf{P}(X) \subset \mathcal{B}_c(X)$. (ii) Let $X$ and $Y$ be Polish spaces and $A \in \mathbf{P}(X \times Y)$. Then, there exists a projective function $\phi : \textup{proj}_X(A) \to Y$ such that $\textup{Graph}(\phi) \subset A$.

Theorems & Definitions (39)

  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 1
  • Definition 3
  • Remark 2
  • Definition 4
  • Theorem 2
  • Definition 5
  • Theorem 3
  • ...and 29 more