Consumption-Investment with anticipative noise
Mario Ayala, Benjamin Vallejo Jiménez
TL;DR
The paper shows that interpreting stochastic noise with a general $\alpha$-integral systematically shifts the effective drift of asset returns, even when the wealth dynamics and self-financing constraint are fixed. For logarithmic utility and constant volatilities, this yields closed-form policy rules where consumption remains a fixed wealth fraction and portfolio weights shift by $\alpha$-dependent terms, with a simple single-asset form $\theta_{\alpha}^{*}=\dfrac{\mu-r}{\sigma^{2}}+\alpha$. When volatility is driven by a factor correlated with returns, the $\alpha$-interpretation induces a state-dependent drift correction proportional to the instantaneous covariation, exemplified by the Heston model where the optimal risky exposure scales inversely with current variance and includes a constant $\alpha$-drift term. The analysis highlights that stochastic-noise conventions are economically meaningful, especially in factor-rich markets, and provides a framework to translate between Itô, Stratonovich, and Klimontovich viewpoints while preserving tractable optimization. Overall, the results clarify how modeling choices for noise interpretation affect intertemporal decisions and carry practical implications for dynamic asset allocation under stochastic volatility.
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.