Consumption-Investment with anticipative noise

TL;DR

This paper extends the classical consumption-investment model by incorporating a general $oldsymbol{ extalpha}$-integral for asset returns, revealing how noise interpretation affects optimal policies and introduces state-dependent effects in stochastic volatility settings.

Contribution

It derives closed-form optimal policies under a general noise interpretation and analyzes state-dependent effects in stochastic volatility models like Heston.

Findings

Optimal portfolio weights are shifted by the noise interpretation parameter $oldsymbol{ extalpha}$.

State-dependent effects emerge when volatility is driven by a stochastic factor.

In the Heston model, risky exposure inversely depends on current variance.

Abstract

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 It\^{o}, 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 $\theta_\alpha^\ast = V^{-1}(\mu-r\mathbf{1})+\alpha\,V^{-1}\operatorname{diag}(V)\mathbf{1}$, 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 $\theta_\alpha^\ast=(\mu-r)/\sigma^{2}+\alpha$. 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.