Empirical Estimates in Economic and Financial Optimization Problems

Many applications from economic and financial practice lead to optimization problems depending on a probability measure. A complete knowledge of the "underlying" measure is a necessary assumption to determine an exact optimal solution and an exact optimal value. Since this condition is not usually fulfilled, the solution is often determined on an empirical data base. Estimates of the optimal value and the optimal solution sets can be obtained by this approach only. Many efforts has been paid to the investigation of the above mentioned estimates. Especially the consistency and the convergence rate have been investigated. However, it was mostly done for "classical" problems and "underlying" distributions with "thin" tails. The aim of this paper is to analyze these estimates from the point of the distribution tails, generally. To this end, first, we recall some known results. Furthermore, we recall stability results based on the Wasserstein metric corresponding to L1 norm (see e.g. Kan (2006b), Kan (2006a)) and employ them to the case of "heavy" tails. Results based on a simulation technique complete our investigation.