Sensitivity Analysis of Nonlinear Behavior with Distorted Probability

One of the important issues in behavioral analysis is that the time consistency property is lost due to the distortion in performance probability. The standard approach in performance optimization, the dynamic programming, fails to work in this area. In this paper, we propose to use an alternative approach, the sensitivity-based analysis, to study this nonlinear behavior with probability distortion. With this approach, we first discover the "mono-linearity" of the distorted performance, which means that after properly changing the underlying probability measure the distorted performance becomes locally linear in probabilities, and the derivative of the distorted performance is simply the expectation, under this new measure, of the sample path based derivative of the performance. The mono-linearity enables us to apply the perturbation analysis to obtain the geometrical property of the distorted performance as a function of the control variables. From the mono-linearity, simulation algorithms for estimating the derivative of distorted performance can be developed, leading to gradient-based search algorithms for the optimal policy. Furthermore, we apply this approach to the portfolio management problem with distorted performance and find a structural property of the price process for the optimal policy, from which we obtain the optimal wealth for the complete market and give several characterizations of optimal solutions in the general market setting of incomplete markets; a solvable incomplete market example with unhedgeable interest rate risk is also provided. We expect that this sensitivity-based approach is generally applicable to optimization in behavioral analysis