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Strategies with minimal norm are optimal for expected utility maximization under high model ambiguity

Laurence Carassus, Johannes Wiesel

Abstract

We investigate an expected utility maximization problem under model uncertainty in a one-period financial market. We capture model uncertainty by replacing the baseline model $\mathbb{P}$ with an adverse choice from a Wasserstein ball of radius $k$ around $\mathbb{P}$ in the space of probability measures and consider the corresponding Wasserstein distributionally robust optimization problem. We show that optimal solutions converge to a strategy with minimal norm when uncertainty is increasingly large, i.e. when the radius $k$ tends to infinity.

Strategies with minimal norm are optimal for expected utility maximization under high model ambiguity

Abstract

We investigate an expected utility maximization problem under model uncertainty in a one-period financial market. We capture model uncertainty by replacing the baseline model with an adverse choice from a Wasserstein ball of radius around in the space of probability measures and consider the corresponding Wasserstein distributionally robust optimization problem. We show that optimal solutions converge to a strategy with minimal norm when uncertainty is increasingly large, i.e. when the radius tends to infinity.
Paper Structure (11 sections, 24 theorems, 136 equations)

This paper contains 11 sections, 24 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Literature review and discussion of Theorem \ref{['thm:main']}
  3. Notation
  4. Financial market
  5. Discussion of Theorem \ref{['thm:main']} and its assumptions
  6. Existence of bounded optimal solutions of $u(x_0)$
  7. Existence of bounded optimal solutions under RAE
  8. Existence of optimal solutions when $U$ is bounded from above
  9. Proof of Theorem \ref{['thm:main']}
  10. Remaining proofs from Section \ref{['sec:discussion']}
  11. Further results on $u(k,w)$

Key Result

Theorem 1.1

Let $U: \mathbb{R} \to \mathbb{R}$ be a non-decreasing, non-constant, differentiable and concave function and assume that there exist some constants $C_1,\underline{x}>0$ such that $U(-\underline{x})<0$ and for all $x\leq - \underline{x}$ Furthermore assume that one of the three conditions holds true: Then $w_k$ is well-defined and $\lim_{k\to \infty} w_k$ exists and belongs to $\mathcal{D}$, whe

Theorems & Definitions (50)

  • Theorem 1.1
  • Remark 1.2
  • Example 1.3
  • Definition 4.1
  • Definition 4.2
  • Proposition 4.3
  • proof
  • Lemma 4.4
  • proof
  • Definition 4.5
  • ...and 40 more