Consumption-Investment with anticipative noise

We revisit the classical Merton consumption–investment problem when risky-asset returns are modeled by stochastic differential equations interpreted through a general $\alpha$-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 ${\theta }_{\alpha }^{\ast }=V^{-1}(\mu -r\mathbf{1})+\alpha V^{-1}\mathrm{diag}(V)\mathbf{1}$, where $V$ is the return covariance matrix and $\mathrm{diag}(V)$ denotes the diagonal matrix with the same diagonal as $V$. In the single-asset case this reduces to ${\theta }_{\alpha }^{\ast }=(\mu -r)\mathrm{/}{\sigma }^2+\alpha$. 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 $\alpha$-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.