Sensitivity analysis of non-linear performance with probability distortion

Human preference over random outcomes may not be as rational as shown in the expected utility theory. Such an "irrational" (as a matter of fact, closer to reality) behavior can be modeled by distorting the probability of the outcomes. Stochastic control of such a distorted performance is difficult because dynamic programming fails to work due to the time inconsistency. In this paper, we formulate the stochastic control problem with the distorted performance and show that the mono-linearity of the distorted performance, which claims that the derivative of the distorted performance equals the expected value of the sample derivative under a changed probability measure, makes the gradient-based sensitivity analysis suitable for optimization of the distorted performance. We derive the first order optimality conditions (or the differential counterpart of the HJK equation) for the optimal solution. We use the portfolio allocation problem in finance as an example of application.