Paper: arXiv 2403.11824

Authors: Laurence Carassus, Massinissa Ferhoune

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.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper operates at an extremely high level of mathematical abstraction, incorporating advanced set-theoretic axioms (Projective Determinacy) to resolve robust utility maximization problems under nonconcave utilities. However, it is purely theoretical with zero empirical validation or backtesting, making it irrelevant for immediate market application but highly significant for mathematical finance theory.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Robust Utility Maximization<br/>under Model Ambiguity & Nonconcave Utility] --> B[Inputs: Discrete-Time Market,<br/>Random/Nonconcave Utility, Projective Priors]
    B --> C[Methodology: Apply Axiom of<br/>Projective Determinacy PD]
    C --> D[Analytical Process: Evaluate Prior-by-Prior<br/>Assumptions & Path-Dependent Projective Graphs]
    D --> E[Key Finding 1: Existence of Optimal Strategy<br/>for Upper-Semicontinuous Utility]
    D --> F[Key Finding 2: Counterexamples Justifying<br/>Necessary Market Assumptions]