Analysis of non-linear behavior - A sensitivity-based approach

One of the important issues in behavioral analysis is that the law of iterated expectation is lost due to the distortion in performance probability. The standard dynamic programming fails to work in this area. In this paper, we propose to use an alternative approach, the sensitivity-based approach, to solve the portfolio management problem in an environment with probability distortion. We show that after changing the underlying probability measure the distorted performance maintains some linearity, and the derivative of the distorted performance is simply the expectation of the sample path based derivative of the performance under this new measure, which can be obtained by perturbation analysis. We also provide simulation algorithms for the derivative of distorted performance. We apply this approach to the initial allocation problem with the distorted performance probability and obtained an optimal policy. We expect that this approach is applicable to other problems in the area of optimization in behavioral analysis.