Relative Growth Rate Optimization Under Behavioral Criterion

This paper studies a continuous-time optimal portfolio selection problem in a complete market for a behavioral investor whose preference is of the prospect type with probability distortion. The investor is concerned with the terminal relative growth rate (log-return) instead of absolute capital value. This model can be regarded as an extension of the classical growth optimal problem to the behavioral framework. It leads to a new type of M-shaped utility maximization problem under nonlinear Choquet expectation. Due to the presence of probability distortion, the classical stochastic control methods are not applicable. Instead, we use the martingale method, concavification, and quantile optimization techniques to derive the closed-form optimal growth rate. We find that the benchmark growth rate has a significant impact on investment behaviors. Compared to S. Zhang, H. Q. Jin, and X. Zhou [Acta Math. Sin. (Engl. Ser.), 27 (2011), pp. 255-274] where the same preference measure is applied to the terminal relative wealth, we find a new phenomenon when the investor's risk tolerance level is high and the market state is bad. In addition, our optimal wealth in every scenario is less sensitive to the pricing kernel and thus more stable than theirs.