Consumption-Investment with anticipative noise

Paper: arXiv 2602.08527

Authors: Mario Ayala, Benjamin Vallejo Jiménez

Abstract

We revisit the classical Merton consumption–investment problem when risky-asset returns are modeled by stochastic differential equations interpreted through a general $α$-integral, interpolating between Itô, Stratonovich, and related conventions. Holding preferences and the investment opportunity set fixed, changing the noise interpretation modifies the effective drift of asset returns in a systematic way. For logarithmic utility and constant volatilities, we derive closed-form optimal policies in a market with $n$ risky assets: optimal consumption remains a fixed fraction of wealth, while optimal portfolio weights are shifted according to $θ_α^\ast = V^{-1}(μ-r\mathbf{1})+α,V^{-1}\operatorname{diag}(V)\mathbf{1}$, where $V$ is the return covariance matrix and $\operatorname{diag}(V)$ denotes the diagonal matrix with the same diagonal as $V$. In the single-asset case this reduces to $θ_α^\ast=(μ-r)/σ^{2}+α$. We then show that genuinely state-dependent effects arise when asset volatility is driven by a stochastic factor correlated with returns. In this setting, the $α$-interpretation generates an additional drift correction proportional to the instantaneous covariation between factor and return noise. As a canonical example, we analyze a Heston stochastic volatility model, where the resulting optimal risky exposure depends inversely on the current variance level.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical exploration of Merton’s portfolio problem under non-Itô stochastic noise conventions (alpha-integrals and Heston stochastic volatility), deriving elegant closed-form results. However, it is entirely analytical with zero empirical testing or financial market data validation.

Research Flowchart

  flowchart TD
    A["Research Goal: Revisit Merton Problem<br/>under General α-Integral Noise"] --> B["Market Model Inputs:<br/>Log Utility, Assets with α-Drift,<br/>Correlated Factor Volatility"]
    
    B --> C["Methodology:<br/>Formulate HJB Equation with<br/>α-Correction Term α·diag(V)"]
    
    C --> D{"Scenario Analysis"}
    
    D -->|"Constant Volatility<br/>(n-Assets)"| E["Derive Closed-Form Policy:<br/>θ* = V⁻¹(μ - r1) + α V⁻¹ diag(V)1"]
    
    D -->|"Stochastic Volatility<br/>(Heston Model)"| F["Compute Covariation Correction<br/>Factor for Return & Volatility Noise"]
    
    E --> G["Key Outcomes:<br/>1. Consumption fraction remains unchanged<br/>2. Constant shift (+α for 1D) in risky weight<br/>3. Heston exposure varies inversely with variance"]
    F --> G

Cited References (37)

1. Reversibility, covariance and coarse-graining for Langevin dynamics: On the choice of multiplicative noise (2025) — link
2. Simulating Portfolio Decisions under Uncertainty When the Risky Asset and Short Rate Are Modulated by an Inhomogeneous and Asset-Dependent Markov Chain (2022) — link
3. On the relation between Stratonovich and Ito integrals with functional integrands of conditional measure flows (2021) — link
4. Algorithmic and High Frequency Trading (2016) — link
5. OPTIMAL CONSUMPTION AND PORTFOLIO DECISIONS WHEN THE RISKY ASSET IS DRIVEN BY A TIME-INHOMOGENEOUS MARKOV MODULATED DIFFUSION PROCESS (2015) — link
6. Pricing under rough volatility (2015) — link
7. Volatility is rough (2014) — link
8. High frequency market microstructure noise estimates and liquidity measures (2009) — link

0 of 37 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.