Numerical Solution of Control Problems under Uncertainty and Perturbation of Input Data with Applications in Finance

In contrast to the stochastic differential equation models used for the calculation of the term structure of interest rates, we develop an approach based on linear dynamical systems under non-stochastic uncertainty with perturbations. The uncertainty is described in terms of unknown feasible sets for varying parameters, whose values are otherwise unknown. Observations are used intrinsically in the approach for obtaining more accurate estimates of the dynamical system. Parameters are estimated by minimizing the maximum absolute value of measurement errors, which is a linear or nonlinear semi-infinite programming problem. A regularized logarithmic barrier method for solving (ill-posed) convex semi-infinite programming problems is suggested. In this method a multistep proximal regularization is coupled with an adaptive discretization strategy in the framework of an interior point approach. A special deleting rule permits to use only a part of the constraints of the discretized problems. Convergence of the method and its stability with respect to data perturbations in the cone of convex C-functions are studied. The strong and permanent change of the structure of the stock exchange in practice needs a short term prediction method, which is constructed on the basis of the solutions of the semi-infinite models for a two years DAX observation. The performance of the forecast model is observed by means of a technical trading system developed for DWS Investment S.A. Luxembourg.