Paper: arXiv 2312.00266

Authors: Weixuan Xia

Abstract

The main objective of this paper is to develop a martingale-type solution to optimal consumption–investment choice problems ([Merton, 1969] and [Merton, 1971]) under time-varying incomplete preferences driven by externalities such as patience, socialization effects, and market volatility. The market is composed of multiple risky assets and multiple consumption goods, while in addition there are multiple fluctuating preference parameters with inexact values connected to imprecise tastes. Utility maximization is a multi-criteria problem with possibly function-valued criteria. To come up with a complete characterization of the solutions, first we motivate and introduce a set-valued stochastic process for the dynamics of multi-utility indices and formulate the optimization problem in a topological vector space. Then, we modify a classical scalarization method allowing for infiniteness and randomness in dimensions and prove results of equivalence to the original problem. Illustrative examples are given to demonstrate practical interests and method applicability progressively. The link between the original problem and a dual problem is also discussed, relatively briefly. Finally, using Malliavin calculus with stochastic geometry, we find optimal investment policies to be generally set-valued, each of whose selectors admits a four-way decomposition involving an additional indecisiveness risk-hedging portfolio. Our results touch on new directions for optimal consumption–investment choices in the presence of incomparability and time inconsistency, also signaling potentially testable assumptions on the variability of asset prices. Simulation techniques for set-valued processes are studied for how solved optimal policies can be computed in practice.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly sophisticated theoretical framework extending Merton’s portfolio problem using set-valued stochastic processes, Malliavin calculus, and topological vector spaces. While heavily innovative and mathematically rigorous, it is purely theoretical with minimal empirical validation beyond conceptual simulations.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Merton Problem under Time-Varying Incomplete Preferences] --> B[Inputs: Multi-asset/goods Market & Set-Valued Preference Parameters]
    B --> C[Formulate Set-Valued Stochastic Process for Multi-Utility Indices]
    C --> D[Apply Modified Scalarization Method in Topological Vector Space]
    D --> E[Derive Dual Problem & Apply Malliavin Calculus with Stochastic Geometry]
    E --> F[Key Outcome: Set-Valued Optimal Policies & 4-Way Decomposition with Indecisiveness Hedging]
    F --> G[Practical Application: Simulation Techniques for Computation & Market Testing]