Monge-Kantorovich transport problems, optimal couplings, and multivariate risk.

Transportation problems and the determination of optimal couplings dates back to Monge 1781 and then to papers of Kantorovich and Rubinstein in the fourties. Lately this problem was handled general form, and it has found applications in analysis, limit theory, and algorithms. A two-volume treatise on transportation problems appeared recently at Springer (Rachev/Rüschendorf). In these books a survey of duality theory of transportation problems and their applications is given. Intensive work is going on on the connection between perfectnes and the validity of duality theorems (Ramachandran/Rüschendorf). It turns out that, in some way, the most general duality theorems can be expected in the context of perfect measure spaces. A currently very interesting application is the modelling and description of dependent risks, especially of the influence of stochastic dependencies on risk measures as for example on the value at risk. Optimal transportation plans could be determined for various concrete problems. Coworker: Jan Bergenthum

Phone: 0761/203-5665
Email: ruschen@stochastik.uni-freiburg.de

Start of project: 2002
End of project: (unlimited)

Albert-Ludwigs-University Freiburg
Prof. Dr. Ludger Rüschendorf
Abteilung für Mathematische Stochastik
Prof. Dr. Ludger Rüschendorf
Ernst-Zermelo-Straße 1
79104 Freiburg
Germany

Phone: 0761/203-5664
Fax: 0761/203-5661
Email: sekretariat@stochastik.uni-freiburg.de
http://www.stochastik.uni-freiburg.de/rueschendorf
Actual Research Report

Monge-Kantorovich Transportproblem, Anwendung auf optimale C

• Rüschendorf L: Stochastic ordering of risks, influence of dependence and a.s. constructions. Preprint Uni Freiburg, 2003; 14.
• Rüschendorf L: Comparison of multivariate risks, Frechet-bounds, and positive dependence. Preprint Uni Freiburg, 2003; 10.
• Ramachandran D, Rüschendorf L: Assignment models for constrained marginals and restricted markets. In: Cuadras et al. (Hrsg.): Probability with Given Marginals. (Proceedings of the Barcelona conference) Kluwer, 2002; 195-209.
• Rüschendorf L, Uckelmann L: Variance minimization and random variables with constant sum. In: Cuadras et al. (Hrsg.): Distributions with Given Marginals. Kluwer, 2002; 211-222.
• Ramachandran D, Rüschendorf L: On the Monge-Kantorovich duality theorem. Theory of Probability and Its Applications, 2000; 45: 350-356.
• Rüschendorf L, Uckelmann L: Numerical and analytical results for the transportation problem of Monge-Kantorovich. Metrika, 2000; 51: 245-258.
• Ramachandran D, Rüschendorf L: An extension of the nonatomic assignment model. In: Alkan A, Aliprantes, Yannelis (Hrsg.): Current Trends in Economics. Theory and Applications.Proceedings of the 3rd international meeting held in Antalya, Turkey, June 1997. Springer, 1999; 405-412.
• Rachev S.T., Rüschendorf L.: Mass Transportation Problems. Vol. I: Theory.Springer, 1998.
• Rachev S.T., Rüschendorf L.: Mass Transportation Problems. Vol. II: Applications.Springer, 1998.
• Uckelmann L.: Über das Monge-Kantorovich Transportproblem und dessen Verallgemeinerungen Dissertation Universität Freiburg, 1998.
• Rüschendorf L, Uckelmann L: On optimal multivariate couplings. In: Benes V, Stepan I (Hrsg.): Proceedings of Prague 1996 conference on marginal problems. Kluwer, 1997; 261-274.
• Uckelmann L.: Optimal couplings between onedimensional distributions. In: Benes V., Stepan I. (Hrsg.): Proceedings of Prague 1996 conference on marginal problems Kluwer, 1997; 275-281.
• Cuesta J, Matran C, Rachev S.T, Rüschendorf L: Mass transportation problems in probability theory. Math. Scientist, 1996; 21: 34-72.
• Ramachandran D, Rüschendorf L: Duality and perfect probability spaces. Trans. Amer. Math. Soc., 1996; 124: 2223-2228.
• Ramachandran D, Rüschendorf L: A general duality theorem for marginal problems. Prob. Theory Rel. Fields, 1995; 101: 311-319.
• Rüschendorf L: Optimal solutions of multivariate coupling problems. Applic. Mathematicae, 1995; 23: 325-338.