The continuous-time pre-commitment KMM problem in incomplete markets

This paper studies the continuous-time pre-commitment KMM problem proposed by Klibanoff, Marinacci and Mukerji (Econometrica 73 (2005) 1849–1892) in incomplete financial markets, which concerns portfolio selection under smooth ambiguity. The KMM model aims to maximize the two-fold utility of terminal wealth, which is not included in the classical subjective utility maximization problem. The DM is uncertain about the dominated priors of the financial market, characterized by a second-order distribution (SOD). This paper first simplifies the problem as a one-fold expected utility problem on the second-order space by constructing the efficient frontier. We establish the bipolar relation and the distorted duality theorem by introducing a novel distorted Legendre transformation. This paper shows the existence and uniqueness of the solution under certain assumptions and provides semi-explicit forms of the optimal terminal wealth and strategy. Explicit forms of optimal strategies are presented for CRRA, CARA and HARA utilities in the case of Gaussian SOD in a Black–Scholes financial market. The results indicate that DMs with higher ambiguity aversion are more concerned about extreme market conditions with a larger bias. Finally, the paper presents numerical comparisons with DMs ignoring ambiguity to illustrate its effects on optimal strategies and value functions.