Consumption-investment under alternative stochastic-integral conventions

Abstract

We study the Merton consumption-investment problem when risky asset dynamics are written under stochastic-integral conventions indexed by $\alpha$, interpolating between the Itô ($\alpha =0$), Stratonovich ($\alpha =1/2$), and Hänggi-Klimontovich ($\alpha =1$) conventions. Transforming the drift jointly with the convention leaves the optimal policy unchanged; holding the drift coefficient numerically fixed changes the investor's opportunity set. Under logarithmic utility, consumption is unaffected while the optimal portfolio weight shifts by $\alpha V^{-1}{d}_V$; for a Heston asset, ${\pi }_t^{\ast }=\alpha +(\mu -r+\alpha \varrho \xi /2)/V_t$, which can change sign across volatility states. Identifying the convention from return data alone is impossible without additional structural restrictions.

arXiv comment39 pages, 2 figures. Substantially revised version