Robust Consumption-Investment with Return Ambiguity: A Dual Approach with Volatility Ambiguity

Kyunghyun Park, Hoi Ying Wong

Abstract: Consider a robust consumption-investment problem for a risk- and ambiguity-averse investor who is concerned about return ambiguity in risky asset prices. When the investor aims to maximize the worst-case scenario of his/her consumption-investment objective, we propose a dual approach to the robust optimization problem in a dual economy with volatility ambiguity. Using the $G$-expectation framework, we establish the duality theorem to bridge between the primal problem with return ambiguity and the dual problem with volatility ambiguity, and hence characterize the robust strategy for a general class of utility functions subject to the nonnegative consumption rate and wealth constraints. The volatility ambiguity in the dual problem induces correlation ambiguity when the primal economy comprises multiple risky assets with return ambiguity. By analyzing the dual economy, we show that the robust investment strategy favors a sparse portfolio, in addition to its usual feature---having the least exposure to ambiguity.