  This paper studies a continuous-time optimal portfolio selection problem in
the complete market for a behavioral investor whose preference is of the
prospect type with probability distortion. The investor concerns about 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. By the martingale method, concavification and quantile optimization
techniques, we derive the closed-form optimal growth rate. We find that the
benchmark growth rate has a significant impact on investment behaviors.
Compared to Zhang et al 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 states are bad. In addition, our optimal
wealth in every scenario is less sensitive to the pricing kernel and thus more
stable than theirs.
