Robust optimal investment and consumption strategies with portfolio constraints and stochastic environment

Paper: arXiv 2407.02831

Abstract

We investigate a continuous-time investment-consumption problem with model uncertainty in a general diffusion-based market with random model coefficients. We assume that a power utility investor is ambiguity-averse, with the preference to robustness captured by the homothetic multiplier robust specification, and the investor’s investment and consumption strategies are constrained to closed convex sets. To solve this constrained robust control problem, we employ the stochastic Hamilton-Jacobi-Bellman-Isaacs equations, backward stochastic differential equations, and bounded mean oscillation martingale theory. Furthermore, we show the investor incurs (non-negative) utility loss, i.e. the loss in welfare, if model uncertainty is ignored. When the model coefficients are deterministic, we establish formally the relationship between the investor’s robustness preference and the robust optimal investment-consumption strategy and the value function, and the impact of investment and consumption constraints on the investor’s robust optimal investment-consumption strategy and value function. Extensive numerical experiments highlight the significant impact of ambiguity aversion, consumption and investment constraints, on the investor’s robust optimal investment-consumption strategy, utility loss, and value function. Key findings include: 1) short-selling restriction always reduces the investor’s utility loss when model uncertainty is ignored; 2) the effect of consumption constraints on utility loss is more delicate and relies on the investor’s risk aversion level.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper employs advanced mathematical tools like stochastic Hamilton-Jacobi-Bellman-Isaacs equations, backward stochastic differential equations, and bounded mean oscillation martingale theory, indicating very high mathematical complexity. However, it lacks empirical data, backtests, or implementation-heavy elements, focusing instead on theoretical derivations and conceptual numerical experiments.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Optimal investment-consumption under model uncertainty with constraints"] --> B["Methodology Setup<br>Stochastic HJB-Isaacs & BSDEs"]
  B --> C["Input Data<br>Stochastic market coefficients & constraint sets"]
  C --> D["Computational Process<br>Apply BMO martingale theory"]
  D --> E["Outcome: Robust Strategies<br>Optimal investment/consumption paths"]
  E --> F["Key Findings<br>Short-selling reduces utility loss<br>Consumption constraints depend on risk aversion"]

Cited References (65)

1. Robust consumption and portfolio policies when asset prices can jump (2019)
2. A quartet of semigroups for model specification, robustness, prices of risk, and model detection (2003)
3. Risks for the long run: A potential resolution of asset pricing puzzles (2004)
4. Introduction to Nonlinear Optimization: Theory, Algorithms, and Applications with MATLAB (2004)
5. Time-consistency of optimal investment under smooth ambiguity (2021)
6. The robust Merton problem of an ambiguity averse investor (2017)
7. Quantifying distributional model risk via optimal transport (2019)
8. Optimal collective investment: an analysis of individual welfare (2010)

0 of 65 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.