2026/02/09 by Mario Ayala, Benjamin Vallejo Jiménez · 1 voice
Economics, Econometrics and Finance · #q-fin.MF
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 θα^∗ = V-1(μ-r1)+α V-1diag(V)1, where V is the return covariance matrix and diag(V) denotes the diagonal matrix with the same diagonal as V. In the single-asset case this reduces to θα^∗=(μ-r)/σ2+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.