Performance-Derivative-Based Optimization

Relative optimization is based on a direct comparison of the performance measures of any two policies. When the two policies under comparison is infinitesimally close, the performance-difference formula becomes the performance-derivative formula. The performance-derivative-based approach is more suitable for non-linear or non-additive performance measure. In this chapter, we derive the first-order optimality conditions (or the differential version of HJB equation). As an example, we study the optimization of a distorted utility measured with distorted probability. We prove that by changing a probability measure the sample derivative is an unbiased estimate of the derivative of the distorted utility. The optimality condition can be derived. This analysis can be applied to study the non-linear behaviour in finance.